Class field theory gives an intrinsic description of all abelian Galois extensions of a given field.
The case of the rational numbers appears very early: the roots of the polynomials X^n ? 1 generate
abelian Galois extensions of the rationals. It is, however, far from obvious that essentially all
abelian extensions of the rationals arise in this way. No analogous explicit description is known
for a general number field. In contrast, over a complete local field such as the p-adic numbers, the
situation is much better. The theory of Lubin–Tate formal groups provides a systematic method
for producing equations whose roots generate all abelian extensions of the field.
Literature
Milne, Class Field Theory; Emily Riehl, Lubin-Tate Formal Groups and Local Class Field Theory -
Bachelor's thesis; Serre, Local Fields
Recommended previous knowledge
basic algebra, including Galois theory. Previous encounter with p-adic numbers is helpful but not
strictly necessary.
Time/Date
Mon 10-12; Wed 12-14; Exercises: Fri 10-12
Location
Mon M101; Wed M103; Exercises: M009
