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Class Field


Given a set P of primes, a field K is called a class field if it is a maximal normal extension of the rationals which splits all of the primes in P, and if P is the maximal set of primes split by K. Here the set P is defined up to the equivalence relation of allowing a finite number of exceptions.

The basic example is the set of primes congruent to 1 (mod 4),

 P={p:p=1 (mod 4)}.

The class field for P is Q(i) because every such prime is expressible as the sum of two squares p=x^2+y^2=(x+iy)(x-iy).


See also

Class Number, Hilbert Class Field, Ideal, Ideal Extension, Local Class Field Theory, Prime Ideal, Unique Factorization

This entry contributed by Todd Rowland

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References

Cohn, H. Introduction to the Construction of Class Fields. New York: Cambridge University Press, 1985.

Referenced on Wolfram|Alpha

Class Field

Cite this as:

Rowland, Todd. "Class Field." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ClassField.html

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