on roots
A theorem which establishes relations between the roots and the coefficients of a polynomial. Let
be a polynomial of degree
with coefficients from some field and with leading coefficient 1. The polynomial
splits over a field containing all the roots of
(e.g. over the splitting field of
, cf. Splitting field of a polynomial) into linear factors:
where
are the roots of
,
. Viète's theorem asserts that the following relations (Viète's formulas) hold:
F. Viète [1] proved this relation for all
, 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) |
A polynomial with leading coefficient
is called monic. Up to sign, the expressions for
in Viète's theorem are nowadays known as the (elementary) symmetric polynomials (of
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