Local Class Field Theory: Some Words before Beginning

The story begins with Hilbert's twelfth problem, that is: how to construct all abelian extensions of a number fields? It is somewhat equivalent to the question: let $K$ be a number field, construct the maximal abelian extension $K^{\text{ab}}$ of $K$. And this question is a starting point of class field theory. In local fields, one can ask for a similar question, let $K$ be a local field, construct the maximal abelian extension $K^{\text{ab}}$ of $K$. This question can be done completely by Lubin-Tate's theory.

Lubin-Tate's theory is based on formal groups, which is a vast and beautiful topic with many applications. Let $R$ be a (commutative) ring, and $F\in R[[X,Y]]$ be a series, then $F$ is said to be a commutative formal group if

1. $F(X,0)=X, F(0,Y)=Y$.
2. $F(X,F(Y,Z))=F(F(X,Y),Z)$ for all $Z\in R[[X,Y]]$.
3. $F(X,Y)=F(Y,X)$.

If one remember about the group structures of elliptic curves over a base field $k$, then addition law on an elliptic curve around the point $\infty$  is exactly a formal group (Silverman, Chapter IV.1). And one can define homomorphism between two formal group. In elliptic curves case, we obtain an isogeny represented around the two infinity points.

Lubin-Tate's formal group is also a specific example of formal groups. It is defined for Forbenius series over a local field. And by attaching suitable things, we obtain an abelian maximal totally ramified extension $K^{\text{ab.tot}}$ of $K$. We also obtain that $Gal(K^{\text{ab.tot}}/K)\cong \mathscr{O}_K^\times$.

Recall that any extension of local field can be decomposed into two parts: totally ramified part and unramified part. For the unramified part, it is somewhat easy, since to obtain an unramified extension of degree $n$, we just need to add $(q^n-1)$ root of unity, where $q$ is the cardinality of the residue field of $K$. And adjoining them all, we obtain $K^{\text{un}}$-the maximal unramified part of $K$. And from this, one has $Gal(K^{\text{un}}/K)\cong \hat{\mathbb{Z}}:=\varprojlim \mathbb{Z}/n\mathbb{Z}$

And now, $K^{\text{ab}}=K^{\text{ab.tot}}K^{\text{un}}$, with $Gal(K^{\text{ab}}/K)=\mathscr{O}_K^\times\times \hat{\mathbb{Z}}$. More than that, with the Artin map, we can also classify abelian extensions of $K$ with given degree.