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Let $F$ be a global field without real places (that is, a function field or a totally imaginary number field). Let $K/F$ be a cyclic extension of degree $n$. Let $S$ be a ${\rm Gal}(K/F)$-invariant ...

It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...

One way to formulate local class field theory is by saying that the local Artin map induces an isomorphism from the profinite completion of $K^\times$ to $\operatorname{Gal}(K^\text{ab}/K)$, which ...

I posted the same question earlier in stack exchange, (https://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field) thinking it is most definitely not a ...

Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...