Artin Reciprocity Law - Finite Extensions of Global Fields

Finite Extensions of Global Fields

The definition of the Artin map for a finite abelian extension L/K of global fields (such as a finite abelian extension of Q) has a concrete description in terms of prime ideals and Frobenius elements.

If is a prime of K then the decomposition groups of primes above are equal in Gal(L/K) since the latter group is abelian. If is unramified in L, then the decomposition group is canonically isomorphic to the Galois group of the extension of residue fields over . There is therefore a canonically defined Frobenius element in Gal(L/K) denoted by or . If Δ denotes the relative discriminant of L/K, the Artin symbol (or Artin map, or (global) reciprocity map) of L/K is defined on the group of prime-to-Δ fractional ideals, by linearity:

\begin{matrix} \left(\frac{L/K}{\cdot}\right):&I_K^\Delta&\longrightarrow&\mathrm{Gal}(L/K)\\ &\displaystyle{\prod_{i=1}^m\mathfrak{p}_i^{n_i}}&\mapsto&\displaystyle{\prod_{i=1}^m\left(\frac{L/K}{\mathfrak{p}_i}\right)^{n_i}.} \end{matrix}

The Artin reciprocity law (or global reciprocity law) states that there is a modulus c of K such that the Artin map induces an isomorphism

where Kc,1 is the ray modulo c, NmL/K is the norm map associated to L/K and is the fractional ideals of L prime to c. Such a modulus c is called a defining modulus for L/K. The smallest defining modulus is called the conductor of L/K and typically denoted .

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