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''on roots''
 
''on roots''
  
A theorem which establishes relations between the roots and the coefficients of a [[Polynomial|polynomial]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966301.png" /> be a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966302.png" /> with coefficients from some field and with leading coefficient 1. The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966303.png" /> splits over a field containing all the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966304.png" /> (e.g. over the splitting field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966305.png" />, cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]) into linear factors:
+
A theorem which establishes relations between the roots and the coefficients of a [[Polynomial|polynomial]]. Let $  f( x) $
 +
be a polynomial of degree $  n $
 +
with coefficients from some field and with leading coefficient 1. The polynomial $  f( x) $
 +
splits over a field containing all the roots of $  f $(
 +
e.g. over the splitting field of $  f( x) $,  
 +
cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]) into linear factors:
  
<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/v/v096/v096630/v0966306.png" /></td> </tr></table>
+
$$
 +
f ( x)  = x  ^ {n} + a _ {n-1} x  ^ {n-1} + \dots + a _ {1} x + a _ {0\ } =
 +
$$
  
<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/v/v096/v096630/v0966307.png" /></td> </tr></table>
+
$$
 +
= \
 +
( x - \alpha _ {1} ) \dots ( x - \alpha _ {n} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966308.png" /> are the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v0966309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v09663010.png" />. Viète's theorem asserts that the following relations (Viète's formulas) hold:
+
where $  \alpha _ {i} $
 +
are the roots of $  f( x) $,  
 +
$  i = 1 \dots n $.  
 +
Viète's theorem asserts that the following relations (Viète's formulas) hold:
  
<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/v/v096/v096630/v09663011.png" /></td> </tr></table>
+
$$
 +
a _ {0}  = (- 1)  ^ {n} \alpha _ {1} \dots \alpha _ {n} ,
 +
$$
  
<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/v/v096/v096630/v09663012.png" /></td> </tr></table>
+
$$
 +
a _ {1}  = (- 1)  ^ {n-1} ( \alpha _ {1} \alpha _ {2} \dots
 +
\alpha _ {n-1} + \alpha _ {1} \dots \alpha _ {n-2} \alpha _ {n} + \dots
 +
$$
  
<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/v/v096/v096630/v09663013.png" /></td> </tr></table>
+
$$
 +
\dots
 +
{} + \alpha _ {2} \alpha _ {3} \dots \alpha _ {n} ),
 +
$$
  
<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/v/v096/v096630/v09663014.png" /></td> </tr></table>
+
$$
 +
\dots \dots \dots \dots
 +
$$
  
<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/v/v096/v096630/v09663015.png" /></td> </tr></table>
+
$$
 +
a _ {n-2}  = \alpha _ {1} \alpha _ {2} + \alpha _ {1} \alpha _ {3} + \dots + \alpha _ {n-1} \alpha _ {n} ,
 +
$$
  
<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/v/v096/v096630/v09663016.png" /></td> </tr></table>
+
$$
 +
a _ {n-1}  = - ( \alpha _ {1} + \dots + \alpha _ {n} ).
 +
$$
  
F. Viète [[#References|[1]]] proved this relation for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v09663017.png" />, but for positive roots only; the general form of Viète's theorem was established by A. Girard [[#References|[2]]].
+
F. Viète [[#References|[1]]] proved this relation for all $  n $,  
 +
but for positive roots only; the general form of Viète's theorem was established by A. Girard [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Viète,  "Opera mathematica"  F. van Schouten (ed.) , Leiden  (1646)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Girard,  "Invention nouvelle en l'algèbre" , Bierens de Haan , Leiden  (1884)  (Reprint)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Viète,  "Opera mathematica"  F. van Schouten (ed.) , Leiden  (1646)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Girard,  "Invention nouvelle en l'algèbre" , Bierens de Haan , Leiden  (1884)  (Reprint)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A polynomial with leading coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v09663018.png" /> is called monic. Up to sign, the expressions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v09663019.png" /> in Viète's theorem are nowadays known as the (elementary) symmetric polynomials (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096630/v09663020.png" /> variables; cf. [[Symmetric polynomial|Symmetric polynomial]]).
+
A polynomial with leading coefficient $  1 $
 +
is called monic. Up to sign, the expressions for $  \alpha _ {i} $
 +
in Viète's theorem are nowadays known as the [[elementary symmetric polynomial]]s (of $  n $
 +
variables; cf. [[Symmetric polynomial|Symmetric polynomial]]).
  
 
Viète's name is sometimes spelled Vièta: Vièta theorem.
 
Viète's name is sometimes spelled Vièta: Vièta theorem.

Latest revision as of 17:59, 16 December 2020


on roots

A theorem which establishes relations between the roots and the coefficients of a polynomial. Let $ f( x) $ be a polynomial of degree $ n $ with coefficients from some field and with leading coefficient 1. The polynomial $ f( x) $ splits over a field containing all the roots of $ f $( e.g. over the splitting field of $ f( x) $, cf. Splitting field of a polynomial) into linear factors:

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

$$ = \ ( x - \alpha _ {1} ) \dots ( x - \alpha _ {n} ), $$

where $ \alpha _ {i} $ are the roots of $ f( x) $, $ i = 1 \dots n $. Viète's theorem asserts that the following relations (Viète's formulas) hold:

$$ a _ {0} = (- 1) ^ {n} \alpha _ {1} \dots \alpha _ {n} , $$

$$ a _ {1} = (- 1) ^ {n-1} ( \alpha _ {1} \alpha _ {2} \dots \alpha _ {n-1} + \alpha _ {1} \dots \alpha _ {n-2} \alpha _ {n} + \dots $$

$$ \dots {} + \alpha _ {2} \alpha _ {3} \dots \alpha _ {n} ), $$

$$ \dots \dots \dots \dots $$

$$ a _ {n-2} = \alpha _ {1} \alpha _ {2} + \alpha _ {1} \alpha _ {3} + \dots + \alpha _ {n-1} \alpha _ {n} , $$

$$ a _ {n-1} = - ( \alpha _ {1} + \dots + \alpha _ {n} ). $$

F. Viète [1] proved this relation for all $ n $, but for positive roots only; the general form of Viète's theorem was established by A. Girard [2].

References

[1] F. Viète, "Opera mathematica" F. van Schouten (ed.) , Leiden (1646)
[2] A. Girard, "Invention nouvelle en l'algèbre" , Bierens de Haan , Leiden (1884) (Reprint)

Comments

A polynomial with leading coefficient $ 1 $ is called monic. Up to sign, the expressions for $ \alpha _ {i} $ in Viète's theorem are nowadays known as the elementary symmetric polynomials (of $ n $ variables; cf. Symmetric polynomial).

Viète's name is sometimes spelled Vièta: Vièta theorem.

References

[a1] B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German)
How to Cite This Entry:
Viète theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem&oldid=23101
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article