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''on roots''

A theorem which establishes relations between the roots and the coefficients of a [[Polynomial|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|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 [[#References|[1]]] proved this relation for all  $  n $, 
but for positive roots only; the general form of Viète's theorem was established by A. Girard [[#References|[2]]].


====Comments====
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 polynomial]]s (of  $  n $
variables; cf. [[Symmetric polynomial|Symmetric polynomial]]).

Viète's name is sometimes spelled Vièta: Vièta theorem.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  F. Viète,   "Opera mathematica"  F. van Schouten (ed.) , Leiden  (1646)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Girard,   "Invention nouvelle en l'algèbre" , Bierens de Haan , Leiden  (1884)  (Reprint)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1''' , Springer  (1967)  (Translated from German)</TD></TR>
</table>