Viète theorem

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)