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Difference between revisions of "Algebraic equation"

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An equation of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114802.png" /> is a [[Polynomial|polynomial]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114803.png" /> in one or more variables (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114804.png" />). An algebraic equation in one variable is an equation of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114805.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114806.png" /> is a non-negative integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114807.png" /> are given, the so-called coefficients of the equation, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114808.png" /> is an unknown which has to be found. It is assumed that the coefficients of the algebraic equation (1) are not all equal to zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a0114809.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148010.png" /> is called the degree of the equation.
+
An equation of the type  $  f _ {n} = 0 $,
 +
where  $  f _ {n} $
 +
is a [[Polynomial|polynomial]] of degree  $  n $
 +
in one or more variables ( $  n \geq  0 $).  
 +
An algebraic equation in one variable is an equation of the form
  
The values of the unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148011.png" /> which satisfy equation (1), i.e. the values which, if substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148012.png" />, will convert this equation into an identity, are known as the roots of the equation (1), or as the roots of the polynomial
+
$$ \tag{1 }
 +
a _ {0} x  ^ {n} + a _ {1} x ^ {n - 1 } + \dots +
 +
a _ {n}  = 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
Here  $  n $
 +
is a non-negative integer,  $  a _ {0} \dots a _ {n} $
 +
are given, the so-called coefficients of the equation, while  $  x $
 +
is an unknown which has to be found. It is assumed that the coefficients of the algebraic equation (1) are not all equal to zero. If  $  a _ {0} \neq 0 $,
 +
then  $  n $
 +
is called the degree of the equation.
 +
 
 +
The values of the unknown  $  x $
 +
which satisfy equation (1), i.e. the values which, if substituted for  $  x $,
 +
will convert this equation into an identity, are known as the roots of the equation (1), or as the roots of the polynomial
 +
 
 +
$$ \tag{2 }
 +
f _ {n} (x) =  a _ {0} x  ^ {n} + a _ {1} x ^ {n - 1 } + \dots +
 +
a _ {n} .
 +
$$
  
 
The roots of a polynomial are related to its coefficients by Viète's formulas (cf. [[Viète theorem|Viète theorem]]). To solve an equation means to find all its roots contained in the range of values of the unknown(s) under consideration.
 
The roots of a polynomial are related to its coefficients by Viète's formulas (cf. [[Viète theorem|Viète theorem]]). To solve an equation means to find all its roots contained in the range of values of the unknown(s) under consideration.
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As far as applications are concerned, the most important case is that of coefficients and roots of an equation that are numbers of a certain kind (e.g. rational, real or complex). The case of the coefficients and roots being elements of an arbitrary [[Field|field]] may also be considered.
 
As far as applications are concerned, the most important case is that of coefficients and roots of an equation that are numbers of a certain kind (e.g. rational, real or complex). The case of the coefficients and roots being elements of an arbitrary [[Field|field]] may also be considered.
  
If a given number (or element of a field) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148014.png" /> is a root of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148015.png" /> then, in accordance with the [[Bezout theorem|Bezout theorem]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148016.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148017.png" /> without remainder. The division may be performed according to the Horner scheme.
+
If a given number (or element of a field) $  c $
 +
is a root of the polynomial $  f _ {n} (x) $
 +
then, in accordance with the [[Bezout theorem|Bezout theorem]], $  f _ {n} (x) $
 +
is divisible by $  x - c $
 +
without remainder. The division may be performed according to the Horner scheme.
  
A number (or element of a field) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148018.png" /> is called a root of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148020.png" /> of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148021.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148022.png" /> is a non-negative integer) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148023.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148024.png" />, but is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148025.png" />. Roots of multiplicity one are called simple roots of the polynomial, other roots are called multiple roots.
+
A number (or element of a field) $  c $
 +
is called a root of multiplicity $  k $
 +
of a polynomial $  f(x) $(
 +
where $  k $
 +
is a non-negative integer) if $  f(x) $
 +
is divisible by $  {(x - c) }  ^ {k} $,  
 +
but is not divisible by $  {(x - c) }  ^ {k+1} $.  
 +
Roots of multiplicity one are called simple roots of the polynomial, other roots are called multiple roots.
  
Each polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148026.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148027.png" /> with coefficients in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148028.png" /> has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148029.png" /> roots in this field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148030.png" />, each root being counted the number of times equal to its multiplicity (consequently, there are not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148031.png" /> different roots).
+
Each polynomial $  f(x) $
 +
of degree $  n > 0 $
 +
with coefficients in a field $  P $
 +
has at most $  n $
 +
roots in this field $  P $,  
 +
each root being counted the number of times equal to its multiplicity (consequently, there are not more than $  n $
 +
different roots).
  
In an [[Algebraically closed field|algebraically closed field]] any polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148032.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148033.png" /> roots (counted according to their multiplicity). In particular, this statement also applies to the field of complex numbers.
+
In an [[Algebraically closed field|algebraically closed field]] any polynomial of degree $  n $
 +
has exactly $  n $
 +
roots (counted according to their multiplicity). In particular, this statement also applies to the field of complex numbers.
  
Equation (1) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148034.png" /> with coefficients from a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148035.png" /> is called irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148036.png" /> if the polynomial (2) is irreducible over this field, i.e. cannot be represented as the product of other polynomials of degrees lower than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148037.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148038.png" />. Otherwise, both the polynomial and the corresponding equation are called reducible. Polynomials of degree zero and zero itself are not considered to be reducible or irreducible. Whether a given polynomial is reducible or irreducible over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148039.png" /> depends on the field in question. Thus, the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148040.png" /> is irreducible over the field of rational numbers, since it has no rational roots, but is reducible over the field of real numbers: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148041.png" />. Similarly, the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148042.png" /> is irreducible over the field of real numbers, but is reducible over the field of complex numbers. Only polynomials of the first degree are irreducible over the field of complex numbers, and any polynomial can be decomposed into linear factors. Only polynomials of the first degree and polynomials of the second degree without real roots are irreducible over the field of real numbers (and all polynomials can be decomposed into products of linear and irreducible quadratic polynomials). Irreducible polynomials of all degrees exist over the field of rational numbers; examples are the polynomials of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148043.png" />. The irreducibility of a polynomial over the field of rational numbers can often be established by Eisenstein's criterion: If, for a polynomial (2) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148044.png" /> with integral coefficients, there exists a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148045.png" /> such that the leading coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148046.png" /> is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148047.png" />, all the remaining coefficients are divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148048.png" />, and the constant term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148049.png" /> is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148050.png" />, then this polynomial is irreducible over the field of rational numbers.
+
Equation (1) of degree $  n $
 +
with coefficients from a field $  P $
 +
is called irreducible over $  P $
 +
if the polynomial (2) is irreducible over this field, i.e. cannot be represented as the product of other polynomials of degrees lower than $  n $
 +
over $  P $.  
 +
Otherwise, both the polynomial and the corresponding equation are called reducible. Polynomials of degree zero and zero itself are not considered to be reducible or irreducible. Whether a given polynomial is reducible or irreducible over a field $  P $
 +
depends on the field in question. Thus, the polynomial $  x  ^ {2} - 2 $
 +
is irreducible over the field of rational numbers, since it has no rational roots, but is reducible over the field of real numbers: $  x  ^ {2} - 2 = (x + \sqrt 2 )(x - \sqrt 2 ) $.  
 +
Similarly, the polynomial $  x  ^ {2} + 1 $
 +
is irreducible over the field of real numbers, but is reducible over the field of complex numbers. Only polynomials of the first degree are irreducible over the field of complex numbers, and any polynomial can be decomposed into linear factors. Only polynomials of the first degree and polynomials of the second degree without real roots are irreducible over the field of real numbers (and all polynomials can be decomposed into products of linear and irreducible quadratic polynomials). Irreducible polynomials of all degrees exist over the field of rational numbers; examples are the polynomials of the form $  x  ^ {n} + 2 $.  
 +
The irreducibility of a polynomial over the field of rational numbers can often be established by Eisenstein's criterion: If, for a polynomial (2) of degree $  n > 0 $
 +
with integral coefficients, there exists a prime number $  p $
 +
such that the leading coefficient $  a _ {0} $
 +
is not divisible by $  p $,  
 +
all the remaining coefficients are divisible by $  p $,  
 +
and the constant term a _ {n} $
 +
is not divisible by $  p  ^ {2} $,  
 +
then this polynomial is irreducible over the field of rational numbers.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148051.png" /> be an arbitrary field. For each polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148052.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148053.png" /> that is irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148054.png" />, there exists an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148055.png" /> containing at least one root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148056.png" /> (cf. [[Extension of a field|Extension of a field]]); moreover, there exists a splitting field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148057.png" />, i.e. a minimal extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148058.png" /> in which this polynomial can be decomposed into linear factors. Every field has an algebraically closed extension.
+
Let $  P $
 +
be an arbitrary field. For each polynomial $  f(x) $
 +
of degree $  n > 1 $
 +
that is irreducible over $  P $,  
 +
there exists an extension of $  P $
 +
containing at least one root of $  f(x) $(
 +
cf. [[Extension of a field|Extension of a field]]); moreover, there exists a splitting field of $  f(x) $,  
 +
i.e. a minimal extension of $  P $
 +
in which this polynomial can be decomposed into linear factors. Every field has an algebraically closed extension.
  
 
==Solvability by radicals.==
 
==Solvability by radicals.==
 
Any algebraic equation of degree not exceeding four is solvable by radicals. Solutions of problems involving special forms of equations of the second and third degrees were already known to ancient Babylonians (2000 B.C.) (cf. [[Quadratic equation|Quadratic equation]]; [[Cubic equation|Cubic equation]]). The theory of the solution of quadratic equations was first expounded in the book Arithmetica by Diophantus (3rd century A.D.). In the 16th century, Italian mathematicians obtained the solution by radicals of equations of the third and fourth degrees with coefficients that are letters (cf. [[Cardano formula|Cardano formula]]; [[Ferrari method|Ferrari method]]). During the 300 years which followed, fruitless efforts were made to find solutions by radicals of equations of the fifth and higher degrees with coefficients that are letters. It was finally demonstrated by N.H. Abel in 1826 that such solutions do not exist.
 
Any algebraic equation of degree not exceeding four is solvable by radicals. Solutions of problems involving special forms of equations of the second and third degrees were already known to ancient Babylonians (2000 B.C.) (cf. [[Quadratic equation|Quadratic equation]]; [[Cubic equation|Cubic equation]]). The theory of the solution of quadratic equations was first expounded in the book Arithmetica by Diophantus (3rd century A.D.). In the 16th century, Italian mathematicians obtained the solution by radicals of equations of the third and fourth degrees with coefficients that are letters (cf. [[Cardano formula|Cardano formula]]; [[Ferrari method|Ferrari method]]). During the 300 years which followed, fruitless efforts were made to find solutions by radicals of equations of the fifth and higher degrees with coefficients that are letters. It was finally demonstrated by N.H. Abel in 1826 that such solutions do not exist.
  
Abel's theorem in modern formulation: Let (1) be an equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148059.png" /> with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148060.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148061.png" /> be an arbitrary field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148062.png" /> be the field of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148063.png" /> with coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148064.png" />; then the roots of equation (1) (lying in some extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148065.png" />) cannot be expressed in terms of the coefficients of this equation by a finite number of operations of addition, subtraction, multiplication and division (operations which are meaningful in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148066.png" />) and root extraction (which is meaningful in the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148067.png" />). In other words, the general equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148068.png" /> is unsolvable by radicals ([[#References|[3]]]).
+
Abel's theorem in modern formulation: Let (1) be an equation of degree $  n > 4 $
 +
with coefficients $  a _ {0} \dots a _ {n} $;  
 +
let $  K $
 +
be an arbitrary field and let $  P $
 +
be the field of rational functions in $  a _ {0} \dots a _ {n} $
 +
with coefficients from $  K $;  
 +
then the roots of equation (1) (lying in some extension of $  P $)  
 +
cannot be expressed in terms of the coefficients of this equation by a finite number of operations of addition, subtraction, multiplication and division (operations which are meaningful in $  P $)  
 +
and root extraction (which is meaningful in the extension of $  P $).  
 +
In other words, the general equation of degree $  n > 4 $
 +
is unsolvable by radicals ([[#References|[3]]]).
  
However, Abel's theorem does not contradict the fact that some algebraic equations with numerical coefficients (or with coefficients from a given field) are solvable by radicals. Some special equations of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148069.png" /> are solvable by radicals (e.g. a [[Two-term equation|two-term equation]]). A complete solution of the problem when an algebraic equation is solvable by radicals was given by E. Galois about the year 1830.
+
However, Abel's theorem does not contradict the fact that some algebraic equations with numerical coefficients (or with coefficients from a given field) are solvable by radicals. Some special equations of degree $  n $
 +
are solvable by radicals (e.g. a [[Two-term equation|two-term equation]]). A complete solution of the problem when an algebraic equation is solvable by radicals was given by E. Galois about the year 1830.
  
The fundamental theorem on the solvability of algebraic equations by radicals in [[Galois theory|Galois theory]] can be stated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148070.png" /> be a polynomial with coefficients from a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148071.png" /> that is irreducible over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148072.png" />. Then, 1) if at least one root of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148073.png" /> can be expressed by radicals in terms of the coefficients of this equation and if the exponents of the radicals are not divisible by the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148074.png" />, then the Galois group of this equation is solvable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148075.png" />; 2) conversely, if the Galois group of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148076.png" /> is solvable over the field and the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148077.png" /> is zero or is higher than all orders of the constituent factors of this group, then all the roots of the equation can be represented by radicals in terms of its coefficients, all exponents of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148078.png" /> involved can be taken to be prime numbers, and the binomial equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148079.png" /> which correspond to these roots can be taken to be irreducible over the fields to which these radicals are adjoined.
+
The fundamental theorem on the solvability of algebraic equations by radicals in [[Galois theory|Galois theory]] can be stated as follows. Let $  f(x) $
 +
be a polynomial with coefficients from a field $  K $
 +
that is irreducible over $  K $.  
 +
Then, 1) if at least one root of the equation $  f(x) = 0 $
 +
can be expressed by radicals in terms of the coefficients of this equation and if the exponents of the radicals are not divisible by the characteristic of $  K $,  
 +
then the Galois group of this equation is solvable over $  K $;  
 +
2) conversely, if the Galois group of the equation $  f(x) = 0 $
 +
is solvable over the field and the characteristic of $  K $
 +
is zero or is higher than all orders of the constituent factors of this group, then all the roots of the equation can be represented by radicals in terms of its coefficients, all exponents of the roots a ^ {1/n} $
 +
involved can be taken to be prime numbers, and the binomial equations $  x  ^ {n} - a = 0 $
 +
which correspond to these roots can be taken to be irreducible over the fields to which these radicals are adjoined.
  
This theorem was proved by Galois for the case in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148080.png" /> is the field of rational numbers; in this case all conditions involving the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148081.png" /> in the above formulation become superfluous.
+
This theorem was proved by Galois for the case in which $  K $
 +
is the field of rational numbers; in this case all conditions involving the characteristic of the field $  K $
 +
in the above formulation become superfluous.
  
Abel's theorem is a consequence of Galois' theorem, since the Galois group of equations of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148082.png" /> with coefficients (that are letters) over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148083.png" /> of rational functions in the coefficients of the equation with coefficients from an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148084.png" />, is the symmetric group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148085.png" />, which is unsolvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148086.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148087.png" /> there exist equations of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148088.png" /> with rational (and integral) coefficients that are unsolvable by radicals. An example of such an equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148089.png" /> is the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148091.png" /> is a prime number. In Galois theory an algebraic equation is solved by reducing it to a chain of simpler equations, which are called resolvents (cf. [[Resolvent|Resolvent]]) of the original equation.
+
Abel's theorem is a consequence of Galois' theorem, since the Galois group of equations of order $  n $
 +
with coefficients (that are letters) over the field $  P $
 +
of rational functions in the coefficients of the equation with coefficients from an arbitrary field $  K $,  
 +
is the symmetric group $  S _ {n} $,  
 +
which is unsolvable for $  n > 4 $.  
 +
For any $  n > 4 $
 +
there exist equations of degree $  n $
 +
with rational (and integral) coefficients that are unsolvable by radicals. An example of such an equation for $  n = 5 $
 +
is the equation $  x  ^ {5} - p  ^ {2} x - p = 0 $,  
 +
where $  p $
 +
is a prime number. In Galois theory an algebraic equation is solved by reducing it to a chain of simpler equations, which are called resolvents (cf. [[Resolvent|Resolvent]]) of the original equation.
  
The solvability of equations by radicals is closely connected with problems involving geometrical constructions with ruler and compasses; in particular, with the problem of the division of the circle into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148092.png" /> equal parts (cf. [[Cyclotomic polynomials|Cyclotomic polynomials]]; [[Primitive root|Primitive root]]).
+
The solvability of equations by radicals is closely connected with problems involving geometrical constructions with ruler and compasses; in particular, with the problem of the division of the circle into $  n $
 +
equal parts (cf. [[Cyclotomic polynomials|Cyclotomic polynomials]]; [[Primitive root|Primitive root]]).
  
 
==Algebraic equations in one unknown with numerical coefficients.==
 
==Algebraic equations in one unknown with numerical coefficients.==
Methods of approximate calculations (e.g. the [[Parabola method|parabola method]]) are generally used to find the roots of algebraic equations of degree higher than two with coefficients from the field of real or complex numbers. It is convenient to begin by getting rid of the multiple roots. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148093.png" /> is a root of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148094.png" /> of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148095.png" /> if and only if the polynomial and its derivatives up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148096.png" />, inclusive, vanish if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148097.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148098.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a01148099.png" /> is divided by the greatest common divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480100.png" /> of this polynomial and its derivative, then the resulting polynomial has the same roots as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480101.png" />, but only of multiplicity one. It is further possible to construct the polynomials whose simple roots are all the roots of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480102.png" /> of equal (given) multiplicity. A polynomial has multiple roots if and only if its [[Discriminant|discriminant]] vanishes.
+
Methods of approximate calculations (e.g. the [[Parabola method|parabola method]]) are generally used to find the roots of algebraic equations of degree higher than two with coefficients from the field of real or complex numbers. It is convenient to begin by getting rid of the multiple roots. A number $  c $
 +
is a root of multiplicity $  k $
 +
of a polynomial $  f(x) $
 +
if and only if the polynomial and its derivatives up to the order $  k - 1 $,  
 +
inclusive, vanish if $  x = c $,  
 +
and if $  f ^ { (k) } (c) \neq 0 $.  
 +
If $  f(x) $
 +
is divided by the greatest common divisor $  d(x) $
 +
of this polynomial and its derivative, then the resulting polynomial has the same roots as $  f(x) $,  
 +
but only of multiplicity one. It is further possible to construct the polynomials whose simple roots are all the roots of the polynomial $  f(x) $
 +
of equal (given) multiplicity. A polynomial has multiple roots if and only if its [[Discriminant|discriminant]] vanishes.
  
 
The determination of the number of roots and bounds on their size are frequently occurring problems. The number
 
The determination of the number of roots and bounds on their size are frequently occurring problems. The number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480103.png" /></td> </tr></table>
+
$$
 +
1 +
 +
\frac{\max _ {i > 0 }  | a _ {i} | }{| a _ {0} | }
 +
 
 +
$$
  
 
can be taken as an upper bound for the modulus of each root (both real and complex) of the algebraic equation (1) with arbitrary complex coefficients. The [[Newton method|Newton method]] usually yields a more exact bound if the coefficients are real. The determination of a lower bound for the positive roots, and of upper and lower bounds for the negative roots, are reduced to the determination of an upper bound for the positive roots.
 
can be taken as an upper bound for the modulus of each root (both real and complex) of the algebraic equation (1) with arbitrary complex coefficients. The [[Newton method|Newton method]] usually yields a more exact bound if the coefficients are real. The determination of a lower bound for the positive roots, and of upper and lower bounds for the negative roots, are reduced to the determination of an upper bound for the positive roots.
  
The simplest method to determine the number of real roots is to use the [[Descartes theorem|Descartes theorem]]. If it is known that all the roots of a given polynomial are real (for example, for the characteristic polynomial of a real symmetric matrix), then Descartes' theorem yields the exact number of roots. By considering the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480104.png" />, the number of negative roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480105.png" /> can be found using the same theorem. The exact number of real roots located in a given interval (in particular, the number of all real roots) of a polynomial with real coefficients without multiple roots can be found by the [[Sturm theorem|Sturm theorem]]. Descartes' theorem is a special case of the [[Budan–Fourier theorem|Budan–Fourier theorem]], which yields an estimate from above for the number of real roots of a polynomial with real coefficients lying in a certain fixed interval.
+
The simplest method to determine the number of real roots is to use the [[Descartes theorem|Descartes theorem]]. If it is known that all the roots of a given polynomial are real (for example, for the characteristic polynomial of a real symmetric matrix), then Descartes' theorem yields the exact number of roots. By considering the polynomial $  f (-x) $,  
 +
the number of negative roots of $  f(x) $
 +
can be found using the same theorem. The exact number of real roots located in a given interval (in particular, the number of all real roots) of a polynomial with real coefficients without multiple roots can be found by the [[Sturm theorem|Sturm theorem]]. Descartes' theorem is a special case of the [[Budan–Fourier theorem|Budan–Fourier theorem]], which yields an estimate from above for the number of real roots of a polynomial with real coefficients lying in a certain fixed interval.
  
 
It is sometimes desirable to find roots of a special type. For instance, Hurwitz' criterion is a necessary and sufficient condition for all the roots of an equation (with complex coefficients) to have negative real parts (cf. [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]).
 
It is sometimes desirable to find roots of a special type. For instance, Hurwitz' criterion is a necessary and sufficient condition for all the roots of an equation (with complex coefficients) to have negative real parts (cf. [[Routh–Hurwitz criterion|Routh–Hurwitz criterion]]).
  
There exists a method for calculating all rational roots of a polynomial with rational coefficients. A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480106.png" /> with rational coefficients has the same roots as the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480107.png" /> with integral coefficients obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480108.png" /> by multiplication by a common multiple of all denominators of the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480109.png" />. The only rational roots of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480111.png" />, with integral coefficients are found among the irreducible fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480112.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480113.png" /> is a divisor of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480115.png" /> is a divisor of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480116.png" /> (and only those fractions among them such that, for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480117.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480118.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480119.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480120.png" />, then all rational roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480121.png" /> (if any) are integers, divisors of the constant term, and can be found by trial and error.
+
There exists a method for calculating all rational roots of a polynomial with rational coefficients. A polynomial $  f (x) $
 +
with rational coefficients has the same roots as the polynomial $  g(x) $
 +
with integral coefficients obtained from $  f(x) $
 +
by multiplication by a common multiple of all denominators of the coefficients of $  f(x) $.  
 +
The only rational roots of a polynomial $  g (x) = b _ {0} x  ^ {n} + b _ {1} x  ^ {n-1} + \dots + b _ {n} $,  
 +
$  b _ {n} \neq 0 $,  
 +
with integral coefficients are found among the irreducible fractions $  p/q $
 +
in which $  p $
 +
is a divisor of the number $  b _ {n} $
 +
and $  q $
 +
is a divisor of the number $  b _ {0} $(
 +
and only those fractions among them such that, for any integer $  m $,  
 +
the number $  g(m) $
 +
is divisible by $  p - mq $).  
 +
If $  b _ {0} = 1 $,  
 +
then all rational roots of $  g(x) $(
 +
if any) are integers, divisors of the constant term, and can be found by trial and error.
  
 
==Systems of algebraic equations.==
 
==Systems of algebraic equations.==
 
Concerning systems of algebraic equations of the first degree, see [[Linear equation|Linear equation]].
 
Concerning systems of algebraic equations of the first degree, see [[Linear equation|Linear equation]].
  
A system of any two algebraic equations of any degree in two unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480123.png" /> may be written as:
+
A system of any two algebraic equations of any degree in two unknowns $  x $
 +
and  $  y $
 +
may be written as:
 +
 
 +
$$ \tag{3 }
 +
\left .
 +
\begin{array}{l}
 +
f ( x , y )  = a _ {0} ( x ) y  ^ {n} + a _ {1} ( x ) y ^
 +
{n - 1 } + \dots + a _ {n} ( x )  = 0 ,  \\
 +
g ( x , y )  = b _ {0} ( x ) y  ^ {s} + b _ {1} ( x ) y ^
 +
{s - 1 } + \dots + b _ {s} ( x )  = 0 ,  \\
 +
\end{array}
 +
\right \}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
where  $  a _ {i} (x), b _ {j} (x) $
 +
are polynomials in one unknown  $  x $.  
 +
If a certain numerical value is assigned to  $  x $,
 +
a system of two equations in one unknown  $  y $
 +
with constant coefficients  $  a _ {i} , b _ {j} $
 +
is obtained. The [[Resultant|resultant]] of this system is the following determinant:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480125.png" /> are polynomials in one unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480126.png" />. If a certain numerical value is assigned to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480127.png" />, a system of two equations in one unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480128.png" /> with constant coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480129.png" /> is obtained. The [[Resultant|resultant]] of this system is the following determinant:
+
$$
 +
R (f, g)  = \
 +
\left |
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480130.png" /></td> </tr></table>
+
\begin{array}{cccccccc}
 +
a _ {0}  &a _ {1}  &\dots  &a _ {n}  & 0  & 0  &\dots  & 0  \\
 +
0  &a _ {0}  &\dots  &\dots  &a _ {n}  &\dots  &\dots  &\dots  \\
 +
\dots  &\dots  &\dots  &\dots  &\dots  & 0  &\dots  & 0  \\
 +
0  & 0  &\dots  &\dots  & 0 &a _ {0}  &\dots  &a _ {n}  \\
 +
b _ {0}  &b _ {1}  &\dots  &b _ {s}  & 0  & 0  &\dots  & 0  \\
 +
0  &b _ {0}  &\dots  &\dots  &b _ {s}  &\dots  &\dots  &\dots  \\
 +
\dots  &\dots  &\dots  &\dots  &\dots  & 0  &\dots  & 0  \\
 +
0  & 0  &\dots  &\dots  & 0  &b _ {0}  &\dots  &b _ {s}  \\
 +
\end{array}
 +
\
 +
\right | .
 +
$$
  
(There are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480131.png" /> rows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480132.png" />'s and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480133.png" /> rows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480134.png" />'s in this determinant.) The following statement is true: A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480135.png" /> is a root of the resultant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480136.png" /> if and only if the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480138.png" /> have a common root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480139.png" /> or if the two leading coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480140.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480141.png" /> are equal to zero.
+
(There are $  s $
 +
rows of a $'
 +
s and $  n $
 +
rows of $  b $'
 +
s in this determinant.) The following statement is true: A number $  x _ {0} $
 +
is a root of the resultant $  R (f, g) $
 +
if and only if the polynomials $  f (x _ {0} , y ) $
 +
and $  g (x _ {0} , y ) $
 +
have a common root $  y _ {0} $
 +
or if the two leading coefficients $  a _ {0} ( x _ {0} ) $
 +
and $  b _ {0} ( x _ {0} ) $
 +
are equal to zero.
  
Thus, in order to solve the system (3) one must find all roots of the resultant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480142.png" />, substitute each one of these roots into the system (3) and find the common roots of these two equations in one unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480143.png" />. One must also find the common roots of the two polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480144.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480145.png" />, substitute them into (3), and verify if the resulting equations in one unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480146.png" /> have common roots. In other words, the solution of a system of two algebraic equations in two unknowns is reduced to the solution of one equation in one unknown and the calculation of the common roots of two equations in one unknown (the common roots of two or more polynomials in one unknown are the roots of their largest common divisor).
+
Thus, in order to solve the system (3) one must find all roots of the resultant $  R (f, g) $,  
 +
substitute each one of these roots into the system (3) and find the common roots of these two equations in one unknown $  y $.  
 +
One must also find the common roots of the two polynomials $  a _ {0} (x) $
 +
and $  b _ {0} (x) $,  
 +
substitute them into (3), and verify if the resulting equations in one unknown $  y $
 +
have common roots. In other words, the solution of a system of two algebraic equations in two unknowns is reduced to the solution of one equation in one unknown and the calculation of the common roots of two equations in one unknown (the common roots of two or more polynomials in one unknown are the roots of their largest common divisor).
  
 
Systems of any number of algebraic equations with any number of unknowns are solved in a similar manner. This task involves complicated calculations. It is connected with the so-called [[Elimination theory|elimination theory]].
 
Systems of any number of algebraic equations with any number of unknowns are solved in a similar manner. This task involves complicated calculations. It is connected with the so-called [[Elimination theory|elimination theory]].
Line 74: Line 261:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.K. Sushkevich,  "Foundations of higher algebra" , Moscow-Leningrad  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.I. Manin,  "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1969)  pp. 205–230  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.P. Domoryad,  , ''Encyclopaedia of elementary mathematics'' , '''2''' , Moscow  (1951)  pp. 313–511  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.K. Sushkevich,  "Foundations of higher algebra" , Moscow-Leningrad  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.I. Manin,  "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1969)  pp. 205–230  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.P. Domoryad,  , ''Encyclopaedia of elementary mathematics'' , '''2''' , Moscow  (1951)  pp. 313–511  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Abel's theorem mentioned above (the general equation of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011480/a011480147.png" /> is unsolvable by radicals) is also called the Abel–Ruffini theorem.
+
Abel's theorem mentioned above (the general equation of degree $  n > 4 $
 +
is unsolvable by radicals) is also called the Abel–Ruffini theorem.
  
 
Good up-to-date textbooks are [[#References|[a1]]] and [[#References|[a2]]].
 
Good up-to-date textbooks are [[#References|[a1]]] and [[#References|[a2]]].

Latest revision as of 16:09, 1 April 2020


An equation of the type $ f _ {n} = 0 $, where $ f _ {n} $ is a polynomial of degree $ n $ in one or more variables ( $ n \geq 0 $). An algebraic equation in one variable is an equation of the form

$$ \tag{1 } a _ {0} x ^ {n} + a _ {1} x ^ {n - 1 } + \dots + a _ {n} = 0 . $$

Here $ n $ is a non-negative integer, $ a _ {0} \dots a _ {n} $ are given, the so-called coefficients of the equation, while $ x $ is an unknown which has to be found. It is assumed that the coefficients of the algebraic equation (1) are not all equal to zero. If $ a _ {0} \neq 0 $, then $ n $ is called the degree of the equation.

The values of the unknown $ x $ which satisfy equation (1), i.e. the values which, if substituted for $ x $, will convert this equation into an identity, are known as the roots of the equation (1), or as the roots of the polynomial

$$ \tag{2 } f _ {n} (x) = a _ {0} x ^ {n} + a _ {1} x ^ {n - 1 } + \dots + a _ {n} . $$

The roots of a polynomial are related to its coefficients by Viète's formulas (cf. Viète theorem). To solve an equation means to find all its roots contained in the range of values of the unknown(s) under consideration.

As far as applications are concerned, the most important case is that of coefficients and roots of an equation that are numbers of a certain kind (e.g. rational, real or complex). The case of the coefficients and roots being elements of an arbitrary field may also be considered.

If a given number (or element of a field) $ c $ is a root of the polynomial $ f _ {n} (x) $ then, in accordance with the Bezout theorem, $ f _ {n} (x) $ is divisible by $ x - c $ without remainder. The division may be performed according to the Horner scheme.

A number (or element of a field) $ c $ is called a root of multiplicity $ k $ of a polynomial $ f(x) $( where $ k $ is a non-negative integer) if $ f(x) $ is divisible by $ {(x - c) } ^ {k} $, but is not divisible by $ {(x - c) } ^ {k+1} $. Roots of multiplicity one are called simple roots of the polynomial, other roots are called multiple roots.

Each polynomial $ f(x) $ of degree $ n > 0 $ with coefficients in a field $ P $ has at most $ n $ roots in this field $ P $, each root being counted the number of times equal to its multiplicity (consequently, there are not more than $ n $ different roots).

In an algebraically closed field any polynomial of degree $ n $ has exactly $ n $ roots (counted according to their multiplicity). In particular, this statement also applies to the field of complex numbers.

Equation (1) of degree $ n $ with coefficients from a field $ P $ is called irreducible over $ P $ if the polynomial (2) is irreducible over this field, i.e. cannot be represented as the product of other polynomials of degrees lower than $ n $ over $ P $. Otherwise, both the polynomial and the corresponding equation are called reducible. Polynomials of degree zero and zero itself are not considered to be reducible or irreducible. Whether a given polynomial is reducible or irreducible over a field $ P $ depends on the field in question. Thus, the polynomial $ x ^ {2} - 2 $ is irreducible over the field of rational numbers, since it has no rational roots, but is reducible over the field of real numbers: $ x ^ {2} - 2 = (x + \sqrt 2 )(x - \sqrt 2 ) $. Similarly, the polynomial $ x ^ {2} + 1 $ is irreducible over the field of real numbers, but is reducible over the field of complex numbers. Only polynomials of the first degree are irreducible over the field of complex numbers, and any polynomial can be decomposed into linear factors. Only polynomials of the first degree and polynomials of the second degree without real roots are irreducible over the field of real numbers (and all polynomials can be decomposed into products of linear and irreducible quadratic polynomials). Irreducible polynomials of all degrees exist over the field of rational numbers; examples are the polynomials of the form $ x ^ {n} + 2 $. The irreducibility of a polynomial over the field of rational numbers can often be established by Eisenstein's criterion: If, for a polynomial (2) of degree $ n > 0 $ with integral coefficients, there exists a prime number $ p $ such that the leading coefficient $ a _ {0} $ is not divisible by $ p $, all the remaining coefficients are divisible by $ p $, and the constant term $ a _ {n} $ is not divisible by $ p ^ {2} $, then this polynomial is irreducible over the field of rational numbers.

Let $ P $ be an arbitrary field. For each polynomial $ f(x) $ of degree $ n > 1 $ that is irreducible over $ P $, there exists an extension of $ P $ containing at least one root of $ f(x) $( cf. Extension of a field); moreover, there exists a splitting field of $ f(x) $, i.e. a minimal extension of $ P $ in which this polynomial can be decomposed into linear factors. Every field has an algebraically closed extension.

Solvability by radicals.

Any algebraic equation of degree not exceeding four is solvable by radicals. Solutions of problems involving special forms of equations of the second and third degrees were already known to ancient Babylonians (2000 B.C.) (cf. Quadratic equation; Cubic equation). The theory of the solution of quadratic equations was first expounded in the book Arithmetica by Diophantus (3rd century A.D.). In the 16th century, Italian mathematicians obtained the solution by radicals of equations of the third and fourth degrees with coefficients that are letters (cf. Cardano formula; Ferrari method). During the 300 years which followed, fruitless efforts were made to find solutions by radicals of equations of the fifth and higher degrees with coefficients that are letters. It was finally demonstrated by N.H. Abel in 1826 that such solutions do not exist.

Abel's theorem in modern formulation: Let (1) be an equation of degree $ n > 4 $ with coefficients $ a _ {0} \dots a _ {n} $; let $ K $ be an arbitrary field and let $ P $ be the field of rational functions in $ a _ {0} \dots a _ {n} $ with coefficients from $ K $; then the roots of equation (1) (lying in some extension of $ P $) cannot be expressed in terms of the coefficients of this equation by a finite number of operations of addition, subtraction, multiplication and division (operations which are meaningful in $ P $) and root extraction (which is meaningful in the extension of $ P $). In other words, the general equation of degree $ n > 4 $ is unsolvable by radicals ([3]).

However, Abel's theorem does not contradict the fact that some algebraic equations with numerical coefficients (or with coefficients from a given field) are solvable by radicals. Some special equations of degree $ n $ are solvable by radicals (e.g. a two-term equation). A complete solution of the problem when an algebraic equation is solvable by radicals was given by E. Galois about the year 1830.

The fundamental theorem on the solvability of algebraic equations by radicals in Galois theory can be stated as follows. Let $ f(x) $ be a polynomial with coefficients from a field $ K $ that is irreducible over $ K $. Then, 1) if at least one root of the equation $ f(x) = 0 $ can be expressed by radicals in terms of the coefficients of this equation and if the exponents of the radicals are not divisible by the characteristic of $ K $, then the Galois group of this equation is solvable over $ K $; 2) conversely, if the Galois group of the equation $ f(x) = 0 $ is solvable over the field and the characteristic of $ K $ is zero or is higher than all orders of the constituent factors of this group, then all the roots of the equation can be represented by radicals in terms of its coefficients, all exponents of the roots $ a ^ {1/n} $ involved can be taken to be prime numbers, and the binomial equations $ x ^ {n} - a = 0 $ which correspond to these roots can be taken to be irreducible over the fields to which these radicals are adjoined.

This theorem was proved by Galois for the case in which $ K $ is the field of rational numbers; in this case all conditions involving the characteristic of the field $ K $ in the above formulation become superfluous.

Abel's theorem is a consequence of Galois' theorem, since the Galois group of equations of order $ n $ with coefficients (that are letters) over the field $ P $ of rational functions in the coefficients of the equation with coefficients from an arbitrary field $ K $, is the symmetric group $ S _ {n} $, which is unsolvable for $ n > 4 $. For any $ n > 4 $ there exist equations of degree $ n $ with rational (and integral) coefficients that are unsolvable by radicals. An example of such an equation for $ n = 5 $ is the equation $ x ^ {5} - p ^ {2} x - p = 0 $, where $ p $ is a prime number. In Galois theory an algebraic equation is solved by reducing it to a chain of simpler equations, which are called resolvents (cf. Resolvent) of the original equation.

The solvability of equations by radicals is closely connected with problems involving geometrical constructions with ruler and compasses; in particular, with the problem of the division of the circle into $ n $ equal parts (cf. Cyclotomic polynomials; Primitive root).

Algebraic equations in one unknown with numerical coefficients.

Methods of approximate calculations (e.g. the parabola method) are generally used to find the roots of algebraic equations of degree higher than two with coefficients from the field of real or complex numbers. It is convenient to begin by getting rid of the multiple roots. A number $ c $ is a root of multiplicity $ k $ of a polynomial $ f(x) $ if and only if the polynomial and its derivatives up to the order $ k - 1 $, inclusive, vanish if $ x = c $, and if $ f ^ { (k) } (c) \neq 0 $. If $ f(x) $ is divided by the greatest common divisor $ d(x) $ of this polynomial and its derivative, then the resulting polynomial has the same roots as $ f(x) $, but only of multiplicity one. It is further possible to construct the polynomials whose simple roots are all the roots of the polynomial $ f(x) $ of equal (given) multiplicity. A polynomial has multiple roots if and only if its discriminant vanishes.

The determination of the number of roots and bounds on their size are frequently occurring problems. The number

$$ 1 + \frac{\max _ {i > 0 } | a _ {i} | }{| a _ {0} | } $$

can be taken as an upper bound for the modulus of each root (both real and complex) of the algebraic equation (1) with arbitrary complex coefficients. The Newton method usually yields a more exact bound if the coefficients are real. The determination of a lower bound for the positive roots, and of upper and lower bounds for the negative roots, are reduced to the determination of an upper bound for the positive roots.

The simplest method to determine the number of real roots is to use the Descartes theorem. If it is known that all the roots of a given polynomial are real (for example, for the characteristic polynomial of a real symmetric matrix), then Descartes' theorem yields the exact number of roots. By considering the polynomial $ f (-x) $, the number of negative roots of $ f(x) $ can be found using the same theorem. The exact number of real roots located in a given interval (in particular, the number of all real roots) of a polynomial with real coefficients without multiple roots can be found by the Sturm theorem. Descartes' theorem is a special case of the Budan–Fourier theorem, which yields an estimate from above for the number of real roots of a polynomial with real coefficients lying in a certain fixed interval.

It is sometimes desirable to find roots of a special type. For instance, Hurwitz' criterion is a necessary and sufficient condition for all the roots of an equation (with complex coefficients) to have negative real parts (cf. Routh–Hurwitz criterion).

There exists a method for calculating all rational roots of a polynomial with rational coefficients. A polynomial $ f (x) $ with rational coefficients has the same roots as the polynomial $ g(x) $ with integral coefficients obtained from $ f(x) $ by multiplication by a common multiple of all denominators of the coefficients of $ f(x) $. The only rational roots of a polynomial $ g (x) = b _ {0} x ^ {n} + b _ {1} x ^ {n-1} + \dots + b _ {n} $, $ b _ {n} \neq 0 $, with integral coefficients are found among the irreducible fractions $ p/q $ in which $ p $ is a divisor of the number $ b _ {n} $ and $ q $ is a divisor of the number $ b _ {0} $( and only those fractions among them such that, for any integer $ m $, the number $ g(m) $ is divisible by $ p - mq $). If $ b _ {0} = 1 $, then all rational roots of $ g(x) $( if any) are integers, divisors of the constant term, and can be found by trial and error.

Systems of algebraic equations.

Concerning systems of algebraic equations of the first degree, see Linear equation.

A system of any two algebraic equations of any degree in two unknowns $ x $ and $ y $ may be written as:

$$ \tag{3 } \left . \begin{array}{l} f ( x , y ) = a _ {0} ( x ) y ^ {n} + a _ {1} ( x ) y ^ {n - 1 } + \dots + a _ {n} ( x ) = 0 , \\ g ( x , y ) = b _ {0} ( x ) y ^ {s} + b _ {1} ( x ) y ^ {s - 1 } + \dots + b _ {s} ( x ) = 0 , \\ \end{array} \right \} $$

where $ a _ {i} (x), b _ {j} (x) $ are polynomials in one unknown $ x $. If a certain numerical value is assigned to $ x $, a system of two equations in one unknown $ y $ with constant coefficients $ a _ {i} , b _ {j} $ is obtained. The resultant of this system is the following determinant:

$$ R (f, g) = \ \left | \begin{array}{cccccccc} a _ {0} &a _ {1} &\dots &a _ {n} & 0 & 0 &\dots & 0 \\ 0 &a _ {0} &\dots &\dots &a _ {n} &\dots &\dots &\dots \\ \dots &\dots &\dots &\dots &\dots & 0 &\dots & 0 \\ 0 & 0 &\dots &\dots & 0 &a _ {0} &\dots &a _ {n} \\ b _ {0} &b _ {1} &\dots &b _ {s} & 0 & 0 &\dots & 0 \\ 0 &b _ {0} &\dots &\dots &b _ {s} &\dots &\dots &\dots \\ \dots &\dots &\dots &\dots &\dots & 0 &\dots & 0 \\ 0 & 0 &\dots &\dots & 0 &b _ {0} &\dots &b _ {s} \\ \end{array} \ \right | . $$

(There are $ s $ rows of $ a $' s and $ n $ rows of $ b $' s in this determinant.) The following statement is true: A number $ x _ {0} $ is a root of the resultant $ R (f, g) $ if and only if the polynomials $ f (x _ {0} , y ) $ and $ g (x _ {0} , y ) $ have a common root $ y _ {0} $ or if the two leading coefficients $ a _ {0} ( x _ {0} ) $ and $ b _ {0} ( x _ {0} ) $ are equal to zero.

Thus, in order to solve the system (3) one must find all roots of the resultant $ R (f, g) $, substitute each one of these roots into the system (3) and find the common roots of these two equations in one unknown $ y $. One must also find the common roots of the two polynomials $ a _ {0} (x) $ and $ b _ {0} (x) $, substitute them into (3), and verify if the resulting equations in one unknown $ y $ have common roots. In other words, the solution of a system of two algebraic equations in two unknowns is reduced to the solution of one equation in one unknown and the calculation of the common roots of two equations in one unknown (the common roots of two or more polynomials in one unknown are the roots of their largest common divisor).

Systems of any number of algebraic equations with any number of unknowns are solved in a similar manner. This task involves complicated calculations. It is connected with the so-called elimination theory.

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (In Russian)
[2] A.K. Sushkevich, "Foundations of higher algebra" , Moscow-Leningrad (1941) (In Russian)
[3] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[4] Yu.I. Manin, "Ueber die Lösbarkeit von Konstruktionsaufgaben mit Zirkel und Lineal" , Enzyklopaedie der Elementarmathematik , 4. Geometrie , Deutsch. Verlag Wissenschaft. (1969) pp. 205–230 (Translated from Russian)
[5] A.P. Domoryad, , Encyclopaedia of elementary mathematics , 2 , Moscow (1951) pp. 313–511 (In Russian)

Comments

Abel's theorem mentioned above (the general equation of degree $ n > 4 $ is unsolvable by radicals) is also called the Abel–Ruffini theorem.

Good up-to-date textbooks are [a1] and [a2].

References

[a1] N. Jacobson, "Lectures in abstract algebra" , 3. Field theory and Galois theory , v. Nostrand (1975)
[a2] N. Jacobson, "Basic algebra" , 1–2 , Freeman (1974–1980)
How to Cite This Entry:
Algebraic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_equation&oldid=45061
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article