Explicit construction of local class field theory via lubin-tate formal groups

Class field theory constitutes a subsection in algebraic number theory which, in particular, investigates Abelian extensions of global fields. On the other hand, local class field theory was introduced by Helmut Hasse and studies the Abelian extensions of local fields with respect to the objects related to the ground field. It was later developed by various important mathematicians such as Schmidt, Chevalley, Nakayama, Artin, Kato and others. There are several approches to local class field theory: Hasse approach, cohomological approach, the explicit methods of Neukirch and Hazewinkel and others. Similar to the theory of complex multiplication on elliptic curves, in their paper in 1965, Lubin and Tate showed that the main theorems in local class field theory can be proved via formal groups over local fields. Using the Lubin-Tate formal groups, they found an explicit way of generating Abelian extensions of local fields. Here, we will study Lubin-Tate theory in detail.