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

Statement of the Kronecker-Weber Theorem 31
Q(ζ 31 )
( ( ))
Q(ζ + ζ 5 + ζ 25 ) Q(ζ + ζ 2 + ζ 4 + ζ 8 + ζ 16 ) Q cos 2π
31
Q( √ −31)
Q(ζ + ζ −1 + ζ 5
+ζ 6 +ζ 25 +ζ 26 )
Q(ζ + ζ −1 + ζ 2 + ζ 4 + ζ 8
+ ζ 15 + ζ 16 + ζ 23 + ζ 27 + ζ 29 )
Q
Figure 4.5: Field Lattice for Q(ζ 31 )
4.6 Statement of the Kronecker-Weber Theorem
Recall at the end of Chapter 3, we defined an abelian extension K/F to be
an extension whose Galois group G(K/F ) was an abelian group. Having
studied the cyclotomic fields, we conclude with this surprising result.
Theorem 20. Every Galois extension K of Q whose Galois group is
abelian is contained in one of the cyclotomic fields Q(ζ n ).
The proof of this theorem is beyond the scope of this thesis, however, L.
Washington presents a proof in his book Introduction to Cyclotomic Fields.