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Viète theorem

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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)


Comments

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://www.encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem&oldid=23172
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article