# Tagged Questions

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global and local fields.

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### Some property of norm residue symbol

On the last page of this article, Artin and Hasse used some property of the norm residue symbol $$\left(\frac{\lambda}{A}\right)_K = \left(\frac{\lambda}{n(A)}\right)_{k_\zeta}$$ (under Beweis von ...
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### Relating the class number of a field, and of its normal closure

Suppose I take a number field $K$, not necessarily Galois, with class number $h_k$ (over $\mathbb{Q}$). Write $\overline{K}$ for the normal closure of $K$. What, if anything, can be said ...
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### Norm map and extension of idele class groups

It's a classic fact in class field theory that for an extension of number fields $L/K$, we have the norm map on idele class groups $\mathcal{N}:C_L\to C_K$ descended from the map on ideles given by ...
I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote: Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian ...
Let $L/K$ be a finite abelian extension of number fields. There is a well defined homomorphism $\Phi: J_K \rightarrow G(L/K)$, called the Artin map, with the following properties (among many): (i) \$\...