
272 Calculating Galois Groups
Part (2) follows from the definition of ∆( f ).
Let G be the Galois group of f , considered as a subgroup of S
n
. If ∆( f ) is a
perfect square in K then δ ∈ K, so δ is fixed by G. Now odd permutations change δ
to −δ , and since char(K) 6= 2 we have δ 6= −δ . Therefore all permutations in G are
even, that is, G ⊆ A
n
. Conversely, if G ⊆ A
n
then δ ∈ G
†
= K. Therefore ∆( f ) is a
perfect square in K.
In order to apply Theorem 22.7, we must calculate ∆( f ) explicitly. Because it is
a symmetric polynomial in the zeros α
j
, it must be given by some polynomial in the
elementary symmetric polynomials s
k
. Brute force calculations show ...