Difference between revisions of "Viète theorem"
Ulf Rehmann (talk | contribs) m (moved Viète theorem to Viete theorem: ascii title) |
Ulf Rehmann (talk | contribs) m (moved Viete theorem to Viète theorem over redirect: revert) |
(No difference)
|
Revision as of 20:20, 25 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=23172