Namespaces
Variants
Actions

Difference between revisions of "Root"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825802.png" />-th root of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825803.png" /> is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825804.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825805.png" />-th power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825806.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825807.png" />.
+
{{TEX|done}}
 +
An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $a$.
  
A root of an algebraic equation over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r0825808.png" />,
+
A root of an algebraic equation over a field $k$,
  
<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/r/r082/r082580/r0825809.png" /></td> </tr></table>
+
$$a_0x^n+\dots+a_{n-1}x+a_n=0,$$
  
is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258010.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258011.png" /> or to an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258012.png" /> (cf. [[Extension of a field|Extension of a field]]) such that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258013.png" /> is substituted for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258014.png" /> the equation becomes an identity. A root of this equation is also called a root or zero of the polynomial
+
is an element $c$ belonging to $k$ or to an extension of $k$ (cf. [[Extension of a field|Extension of a field]]) such that when $c$ is substituted for $x$ the equation becomes an identity. A root of this equation is also called a root or zero of the polynomial
  
<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/r/r082/r082580/r08258015.png" /></td> </tr></table>
+
$$f(x)=a_0x^n+\dots+a_{n-1}x+a_n.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258016.png" /> is a root of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258018.png" /> is divisible (without remainder) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258019.png" /> (see [[Bezout theorem|Bezout theorem]]). Every polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258020.png" /> with real or complex coefficients has at least one root (hence as many roots as its degree, counting multiplicities). The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258021.png" /> may be expressed as a product
+
If $c$ is a root of a polynomial $f(x)$, then $f(x)$ is divisible (without remainder) by $x-c$ (see [[Bezout theorem|Bezout theorem]]). Every polynomial $f(x)$ with real or complex coefficients has at least one root (hence as many roots as its degree, counting multiplicities). The polynomial $f(x)$ may be expressed as a product
  
<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/r/r082/r082580/r08258022.png" /></td> </tr></table>
+
$$f(x)=a_0(x-c_1)\dots(x-c_n),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258023.png" /> are its roots. If some of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258025.png" /> are equal, their common value is called a multiple root (if a root occurs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258026.png" /> times, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258027.png" /> is called the multiplicity of that root).
+
where $c_1,\dots,c_n$ are its roots. If some of the roots $c_1,\dots,c_n$ of $f(x)$ are equal, their common value is called a multiple root (if a root occurs $m$ times, $m$ is called the multiplicity of that root).
  
A root of unity is an element of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258028.png" /> satisfying the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258029.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258030.png" />. The roots of unity form a subgroup of the multiplicative group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258031.png" />. Conversely, all elements of any finite subgroup of the multiplicative group of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258032.png" /> are roots of unity (cf. [[Fermat little theorem|Fermat little theorem]]) and the subgroup itself is cyclic. This is true, in particular, for the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258033.png" /> of all roots of unity of a given degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258034.png" /> contained in the algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258036.png" />, i.e. the subgroup of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258037.png" /> satisfying the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258039.png" /> is relatively prime to the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258040.png" /> (or if the characteristic is 0), then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258041.png" /> is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258042.png" /> and its generators are known as primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258044.png" />-th roots of unity. The number of such roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258045.png" /> is given by the [[Euler function|Euler function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258046.png" />, i.e. the number of residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258047.png" /> which are relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258048.png" />. In a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258049.png" /> there are no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258050.png" />-th roots of unity other than 1.
+
A root of unity is an element of a field $k$ satisfying the equation $x^m=1$ for some natural number $m$. The roots of unity form a subgroup of the multiplicative group of $k$. Conversely, all elements of any finite subgroup of the multiplicative group of a field $k$ are roots of unity (cf. [[Fermat little theorem|Fermat little theorem]]) and the subgroup itself is cyclic. This is true, in particular, for the subgroup $U_n$ of all roots of unity of a given degree $n$ contained in the algebraic closure $\bar k$ of $k$, i.e. the subgroup of all $\zeta\in\bar k$ satisfying the equation $\zeta^n=1$. If $n$ is relatively prime to the characteristic of $k$ (or if the characteristic is 0), then the group $U_n$ is of order $n$ and its generators are known as primitive $n$-th roots of unity. The number of such roots in $U_n$ is given by the [[Euler function|Euler function]] $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are relatively prime to $n$. In a field of characteristic $p>0$ there are no $p$-th roots of unity other than 1.
  
If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258051.png" /> is finitely generated over its prime subfield, then the number of roots of unity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258052.png" /> is finite.
+
If the field $k$ is finitely generated over its prime subfield, then the number of roots of unity in $k$ is finite.
  
In the field of complex numbers, a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258053.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258054.png" />-th root of unity if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258058.png" /> are integers, i.e. if and only if
+
In the field of complex numbers, a number $z$ is an $n$-th root of unity if and only if $|z|=1$ and $\arg z=2\pi m/n$, where $m$ and $n$ are integers, i.e. if and only if
  
<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/r/r082/r082580/r08258059.png" /></td> </tr></table>
+
$$z=e^{2\pi im/n}=\cos\frac{2\pi m}{n}+i\sin\frac{2\pi m}{n};$$
  
in this case the primitive roots of unity are exactly those for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258060.png" />. In the complex plane, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258061.png" />-th roots of unity coincide with the vertices of the regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082580/r08258062.png" />-gon inscribed in the unit circle; this explains the connection of roots of unity with the problem of squaring the circle (construction of polygons, cf. [[Geometric constructions|Geometric constructions]]).
+
in this case the primitive roots of unity are exactly those for which $(m,n)=1$. In the complex plane, the $n$-th roots of unity coincide with the vertices of the regular $n$-gon inscribed in the unit circle; this explains the connection of roots of unity with the problem of squaring the circle (construction of polygons, cf. [[Geometric constructions|Geometric constructions]]).
  
 
Roots of unity appear in number theory as the values of various important number-theoretical functions (Abelian numerical characters; [[Legendre symbol|Legendre symbol]]; [[Möbius function|Möbius function]]; [[Norm-residue symbol|norm-residue symbol]]; etc.). In field theory and algebraic number theory an important position is occupied by fields obtained by adjunction of roots of unity to some ground field (see [[Cyclotomic field|Cyclotomic field]]; [[Cyclotomic extension|Cyclotomic extension]]; [[Kummer extension|Kummer extension]]).
 
Roots of unity appear in number theory as the values of various important number-theoretical functions (Abelian numerical characters; [[Legendre symbol|Legendre symbol]]; [[Möbius function|Möbius function]]; [[Norm-residue symbol|norm-residue symbol]]; etc.). In field theory and algebraic number theory an important position is occupied by fields obtained by adjunction of roots of unity to some ground field (see [[Cyclotomic field|Cyclotomic field]]; [[Cyclotomic extension|Cyclotomic extension]]; [[Kummer extension|Kummer extension]]).

Latest revision as of 13:40, 30 December 2018

An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $a$.

A root of an algebraic equation over a field $k$,

$$a_0x^n+\dots+a_{n-1}x+a_n=0,$$

is an element $c$ belonging to $k$ or to an extension of $k$ (cf. Extension of a field) such that when $c$ is substituted for $x$ the equation becomes an identity. A root of this equation is also called a root or zero of the polynomial

$$f(x)=a_0x^n+\dots+a_{n-1}x+a_n.$$

If $c$ is a root of a polynomial $f(x)$, then $f(x)$ is divisible (without remainder) by $x-c$ (see Bezout theorem). Every polynomial $f(x)$ with real or complex coefficients has at least one root (hence as many roots as its degree, counting multiplicities). The polynomial $f(x)$ may be expressed as a product

$$f(x)=a_0(x-c_1)\dots(x-c_n),$$

where $c_1,\dots,c_n$ are its roots. If some of the roots $c_1,\dots,c_n$ of $f(x)$ are equal, their common value is called a multiple root (if a root occurs $m$ times, $m$ is called the multiplicity of that root).

A root of unity is an element of a field $k$ satisfying the equation $x^m=1$ for some natural number $m$. The roots of unity form a subgroup of the multiplicative group of $k$. Conversely, all elements of any finite subgroup of the multiplicative group of a field $k$ are roots of unity (cf. Fermat little theorem) and the subgroup itself is cyclic. This is true, in particular, for the subgroup $U_n$ of all roots of unity of a given degree $n$ contained in the algebraic closure $\bar k$ of $k$, i.e. the subgroup of all $\zeta\in\bar k$ satisfying the equation $\zeta^n=1$. If $n$ is relatively prime to the characteristic of $k$ (or if the characteristic is 0), then the group $U_n$ is of order $n$ and its generators are known as primitive $n$-th roots of unity. The number of such roots in $U_n$ is given by the Euler function $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are relatively prime to $n$. In a field of characteristic $p>0$ there are no $p$-th roots of unity other than 1.

If the field $k$ is finitely generated over its prime subfield, then the number of roots of unity in $k$ is finite.

In the field of complex numbers, a number $z$ is an $n$-th root of unity if and only if $|z|=1$ and $\arg z=2\pi m/n$, where $m$ and $n$ are integers, i.e. if and only if

$$z=e^{2\pi im/n}=\cos\frac{2\pi m}{n}+i\sin\frac{2\pi m}{n};$$

in this case the primitive roots of unity are exactly those for which $(m,n)=1$. In the complex plane, the $n$-th roots of unity coincide with the vertices of the regular $n$-gon inscribed in the unit circle; this explains the connection of roots of unity with the problem of squaring the circle (construction of polygons, cf. Geometric constructions).

Roots of unity appear in number theory as the values of various important number-theoretical functions (Abelian numerical characters; Legendre symbol; Möbius function; norm-residue symbol; etc.). In field theory and algebraic number theory an important position is occupied by fields obtained by adjunction of roots of unity to some ground field (see Cyclotomic field; Cyclotomic extension; Kummer extension).

References

[1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2] S. Lang, "Algebra" , Addison-Wesley (1984)


Comments

For the concept of a root in Lie algebra theory see Lie algebra, semi-simple and Root system.

How to Cite This Entry:
Root. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Root&oldid=17528
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article