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Difference between revisions of "Viète theorem"

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$$  
 
$$  
f ( x)  =  x  ^ {n} + a _ {n-} 1 x  ^ {n-} 1 + \dots + a _ {1} x + a _ {0\ } =
+
f ( x)  =  x  ^ {n} + a _ {n-1} x  ^ {n-1} + \dots + a _ {1} x + a _ {0\ } =
 
$$
 
$$
  
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$$  
 
$$  
a _ {1}  =  (- 1)  ^ {n-} 1 ( \alpha _ {1} \alpha _ {2} \dots
+
a _ {1}  =  (- 1)  ^ {n-1} ( \alpha _ {1} \alpha _ {2} \dots
\alpha _ {n-} 1 + \alpha _ {1} \dots \alpha _ {n-} 2 \alpha _ {n} + \dots
+
\alpha _ {n-1} + \alpha _ {1} \dots \alpha _ {n-2} \alpha _ {n} + \dots
 
$$
 
$$
  
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$$  
 
$$  
a _ {n-} 2 =  \alpha _ {1} \alpha _ {2} + \alpha _ {1} \alpha _ {3} + \dots + \alpha _ {n-} 1 \alpha _ {n} ,
+
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} ).
+
a _ {n-1}  =  - ( \alpha _ {1} + \dots + \alpha _ {n} ).
 
$$
 
$$
  
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A polynomial with leading coefficient  $  1 $
 
A polynomial with leading coefficient  $  1 $
 
is called monic. Up to sign, the expressions for  $  \alpha _ {i} $
 
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 $
+
in Viète's theorem are nowadays known as the [[elementary symmetric polynomial]]s (of  $  n $
 
variables; cf. [[Symmetric polynomial|Symmetric polynomial]]).
 
variables; cf. [[Symmetric polynomial|Symmetric polynomial]]).
  

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=49154
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article