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

20 Cyclotomic Field Extensions
a = 2 is a generator for G:
a = 2 a 6 = 7 a 11 = 15 a 16 = 5
a 2 = 4 a 7 = 14 a 12 = 11 a 17 = 10
a 3 = 8 a 8 = 9 a 13 = 3 a 18 = 1
a 4 = 16 a 9 = 18 a 14 = 6 a 19 = 2
a 5 = 13 a 10 = 17 a 15 = 12
Table 4.1: Field Generator for Q(ζ 19 )
a 9 is a generator: a 6 is a generator: a 3 is a generator: a 4 is a generator:
a 9 = 18 a 6 = 7 a 3 = 8 a 4 = 16
a 18 = 1 a 12 = 11 a 6 = 7 a 8 = 9
a 18 = 1 a 9 = 18 a 12 = 11
a 12 = 11 a 16 = 5
a 15 = 12 a 2 = 4
a 18 = 1 a 6 = 7
a 10 = 17
a 14 = 6
a 18 = 1
Table 4.2: Subfield Generators
Let us name the above subgroups as follows. Since a 9 a subgroup of
order 2, we will denote this subgroup of C 18 as C 2 . Similarly, a 6 generates
a subgroup of order 3, denoted C 3 . Next, a 3 generates a subgroup of order
6, so we will denote this subgroup C 6 . Finally, a 4 generates a subgroup of
order 9, so this subgroup will be denoted C 9 .
The subgroups of the cyclic group C 18 correspond to H 2 , H 3 , H 6 , and
H 9 , which are subgroups of the group G with powers of σ as generators.
For example, C 2 = {e, a 9 } corresponds to H 2 = {id, σ 9 }.
Figure 4.1 gives a lattice representation of the subgroups. The lines in-