Galois Theory: A Study of Cyclotomic Field ... - Scripps College

Real Subfields and Quadratic Extensions 27
Thus,
∏
f ′ (α i ) = ∏ i − α j ) = ±D.
i
i≠j(α
When we apply this lemma to the polynomial x 19 − 1, its derivative is
19x 18 . The discriminant is given by
□
±D =
18∏
i=0
19ζ i(18) = ζ N 19 19 ,
where the exponent N is some integer. To determine ζ N , we note that ±D
is a rational number, because x p − 1 ∈ Q[x] are rational. The reason D
is rational is because it is in the splitting field K over Q. Because D was
symmetrically defined, it is clearly in the fixed field of the whole Galois
group. That is, any Q-automorphism of K must permute the roots of f;
any permutation of roots will fix D. However, the fixed field K G where G
is the whole Galois group G(K/Q) is itself just Q. Thus, D is in Q. The only
power of ζ which is rational is 1. So, ζ N = 1 and
±D = 19 19 .
The sign of the discriminant is determined to be negative by carefully
comparing the definition of the derivitive and Equation 4.1. So the square
root of this discriminant is δ = √ −19 19 . By definition of D, it is clear that δ
in the field Q(ζ). Since the square factors can be pulled out of a square root,