Lubin-Tate Formal Groups and Local Class Field Theory

The goal of local class field theory is to classify abelian Galois extensions of a local field K. Several definitions of local fields are in use. In this thesis, local fields, which will be defined explicitly in Section 2, are fields that are complete with respect to a discrete valuation and have a finite residue field. A prototypical first example is Qp, the completion of Q with respect to the absolute value | b |p := p−e, where e is the unique integer such that ab = p e c d and p does not divide the product cd. Why study the abelian extensions of local fields? Initially, this area of study was motivated by questions about number fields. When K is a number field, its fractional ideals form a free abelian group generated by the prime ideals, and thus the quotient of this group modulo its principal ideals, called the class group CK , is abelian. There exists a canonical everywhere unramified extension L/K such that the primes that split in L are precisely the principal ideals in K and such that Gal(L/K) ' CK , so the study of class groups leads naturally to the study of abelian extensions of K. In some sense, number fields can be studied by investigating their behavior near a fixed prime ideal p. Let K be a number field with ring of integers A. The ideal p is maximal in A because the quotient A/p is a finite integral domain and hence a field. The inverse limit