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Artin Symbol


Given a number field K, a Galois extension field L, and prime ideals p of K and P of L unramified over p, there exists a unique element sigma=((L/K),P) of the Galois group G=Gal(L/K) such that for every element alpha of L,

 sigma(alpha)=alpha^(N(p)) (mod P),
(1)

where N(p) is the norm of the prime ideal p in K.

The symbol ((L/K),P) is called an Artin symbol. If L is an Abelian extension of K, the Artin symbol ((L/K),P) depends only on the prime ideal p of K lying under P, so it may be written as ((L/K),p). In this case, the Artin symbol can be generalized as follows. Let a be an ideal of K with prime factorization

 a=product_(i=1)^rp_i^(e_i).
(2)

Then the Artin symbol ((L/K),a) is defined by

 ((L/K),a)=product_(i=1)^r((L/K),p_i)^(e_i).
(3)

See also

Artin Map, Class Field, Class Field Theory, Hilbert Class Field

This entry contributed by David Terr

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References

Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1997.

Referenced on Wolfram|Alpha

Artin Symbol

Cite this as:

Terr, David. "Artin Symbol." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ArtinSymbol.html

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