Diet Galois 117
where the s
j
are the elementary symmetric polynomials
s
1
= t
1
+ ···+t
n
s
2
= t
1
t
2
+t
1
t
3
+ ···+t
n−1
t
n
...
s
n
= t
1
...t
n
Here s
r
is the sum of all products of r distinct t
j
.
The symmetric group S
n
acts as symmetries of C(t
1
,...,t
n
):
σ f (t
1
,...,t
n
) = f (t
σ(1)
,...,t
σ(n)
)
for f ∈ C(t
1
,...,t
n
). The fixed field K of S
n
consists, by definition, of all symmetric
rational functions in the t
j
, which is known to be generated over C by the n elemen-
tary symmetric polynomials in the t
j
. That is, K = C(s
1
,...,s
n
). Moreover, the s
j
satisfy no nontrivial polynomial relation: they are independent. There is a classical
proof of these facts based on induction, using ‘symmetrised monomials’
t
a
1
1
t
a
2
2
···t
a
n
n
+ all permutations thereof
and the so-called ‘lexicographic ordering’ ...