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].
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
[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) Zbl 16.0015.01 |
[a1] | B.L. van der Waerden, "Algebra" , 1 , Springer (1967) (Translated from German) |
Viète theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem&oldid=53039