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

28 Cyclotomic Field Extensions
that is, √ −19 19 = 19 9√ −19, so we see
Q(δ) = Q( √ −19).
This gives us a quadratic subfield of Q(ζ). However, L is the only quadratic
subfield, which means Q(δ) = L.
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Note that the above theorem can be applied to any odd prime.
Applying this theorem to Q(ζ 19 ), we can represent our unique quadratic
extension of Q as:
Q(ζ + ζ 4 + ζ 5 + ζ 6 + ζ 7 + ζ 9 + ζ 11 + ζ 16 + ζ 17 ) = Q( √ −19)
Then if we re-visit our field lattice extension, we see
Notice that L quad is not contained in L max . This corresponds to the fact
that (p − 1)/2 is odd. Had it been even, then our discriminant D would
have been a positive value, making δ real. Then, L quad would have to be a
subfield of L max .
The reader might have noticed that the prime 19 was of the form 19 =
2·3 2 +1, or rather twice the square of a prime plus one. Let us now see how
such a prime differs from a prime of the form 2 · p 1 · p 2 + 1, or twice two
unique primes plus one. We will briefly explore the field extension Q(ζ 31 )
over Q.
Example. The subfields will correspond to the subgroups of (Z/31Z) × .
The generator of this group is a = 3 and the generator for this cyclic group
then the automorphism σ = σ 3 which maps ζ 31 to ζ31 3 . The subgroups of G