Class Field Theory

Take K a number field and m a divisor of K. A congruence subgroup H is defined as a subgroup of the group of all fractional ideals relative prime to m (I_K^m) that contains all principal ideals that are generated by elements of K that are equal to 1 (mod m). These principal ideals split completely in all Abelian extensions and are consequently part of the kernel of the Artin map for each Abelian extension L/K.

When there exists an Abelian extension L/K such that m contains all the primes that ramify in L/K and such that H equals the kernel of the Artin map, then L is called the class field of H.

To formulate the main theorems, the equivalence relation on congruence subgroups is needed, namely that H and H^' are called equivalent if there exists a divisor n such that H intersection I_K^n=H^' intersection I_K^n.

Class field theory consists of two basic theorems. The existence theorem states that to every equivalence class of congruence subgroups, there belongs a class field L. The classification theorem states that for each number field K, there is a unique one-to-one correspondence between the Abelian extensions L/K and the equivalence classes of congruence subgroups H.

This is important because this means that all Abelian extensions of a number field can be found using a property that is completely determined within the number field itself.

See also

Class Field, Class Number, Reciprocity Theorem

This entry contributed by Dirk Trappers

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Garbanati, D. "Class Field Theory Summarized." Rocky Mtn. J. Math. 11, 195-225, 1981.Hazewinkel, M. "Local Class Field Theory is Easy." Adv. Math. 18, 148-181, 1975.

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Class Field Theory

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Trappers, Dirk. "Class Field Theory." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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