# Viète theorem

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)