# Viète theorem

on roots

A theorem which establishes relations between the roots and the coefficients of a polynomial. Let $f( x)$ be a polynomial of degree $n$ with coefficients from some field and with leading coefficient 1. The polynomial $f( x)$ splits over a field containing all the roots of $f$( e.g. over the splitting field of $f( x)$, cf. Splitting field of a polynomial) into linear factors:

$$f ( x) = x ^ {n} + a _ {n-1} x ^ {n-1} + \dots + a _ {1} x + a _ {0\ } =$$

$$= \ ( x - \alpha _ {1} ) \dots ( x - \alpha _ {n} ),$$

where $\alpha _ {i}$ are the roots of $f( x)$, $i = 1 \dots n$. Viète's theorem asserts that the following relations (Viète's formulas) hold:

$$a _ {0} = (- 1) ^ {n} \alpha _ {1} \dots \alpha _ {n} ,$$

$$a _ {1} = (- 1) ^ {n-1} ( \alpha _ {1} \alpha _ {2} \dots \alpha _ {n-1} + \alpha _ {1} \dots \alpha _ {n-2} \alpha _ {n} + \dots$$

$$\dots {} + \alpha _ {2} \alpha _ {3} \dots \alpha _ {n} ),$$

$$\dots \dots \dots \dots$$

$$a _ {n-2} = \alpha _ {1} \alpha _ {2} + \alpha _ {1} \alpha _ {3} + \dots + \alpha _ {n-1} \alpha _ {n} ,$$

$$a _ {n-1} = - ( \alpha _ {1} + \dots + \alpha _ {n} ).$$

F. Viète  proved this relation for all $n$, but for positive roots only; the general form of Viète's theorem was established by A. Girard .

A polynomial with leading coefficient $1$ is called monic. Up to sign, the expressions for $\alpha _ {i}$ in Viète's theorem are nowadays known as the elementary symmetric polynomials (of $n$ variables; cf. Symmetric polynomial).