
210 The General Polynomial Equation
Clearly distinct elements of S
n
give rise to distinct K-automorphisms.
The fixed field F of S
n
obviously contains all the symmetric polynomials in the t
i
,
and in particular the elementary symmetric polynomials s
r
= s
r
(t
1
,...,t
n
). We show
that these generate F.
Lemma 18.11. With the above notation, F = K(s
1
,...,s
n
). Moreover,
[K(t
1
,...,t
n
) : K(s
1
,...,s
n
)] = n! (18.2)
Proof. Clearly L = K(t
1
,...,t
n
) is a splitting field of f (t) over both K(s
1
,...,s
n
) and
the possibly larger field F. Since S
n
fixes both of these fields, the Galois group of each
extension contains S
n
, so must equal S
n
. Therefore the fields F and K(s
1
,...,s
n
) are ...