Difference between revisions of "Viète theorem"
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Revision as of 18:54, 24 March 2012
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) 
Viète theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vi%C3%A8te_theorem&oldid=14535