Algebraic Number Theory pp 197-212 | Cite as

# The Artin Symbol, Reciprocity Law, and Class Field Theory

Chapter

## Abstract

Let

*K/k*be an abelian extension, and let p be a prime of*k*which is unramified in*K*. We had seen in Chapter I, §5 that there exists a unique element σ of the Galois group (*G*, lying in the decomposition group*G*_{P}(for any P|p, they all coincide in the abelian case) having the effect$$\sigma \alpha \equiv \alpha ^{\text{Np}} (\bmod \,\,\,\text{P}),\quad \quad \quad \quad \quad \quad \quad \quad \alpha \in o_k .$$

## Keywords

Prime Number Galois Group Finite Index Class Field Open Subgroup
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1986