# Abelian field

## Abelian field

[ə′bēl·yən ′fēld]
(mathematics)
A set of elements a, b, c, … forming Abelian groups with addition and multiplication as group operations where a (b + c) = ab + ac. Also known as Abelian domain; domain.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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11), a real abelian field F satisfies ([H.sub.p]) if and only if F = Q([square root of 5]) or Q([square root of (13)]) (resp.
Denote by [h.sup.M] the class number of a number field M, and by [h.sup.-.sub.M] the relative class number when M is an imaginary abelian field. We set [h.sup.-.sub.p] = [h.sup.-.sub.M] when M = Q([[zeta].sub.p]).
At present, we have no example of an abelian field F which satisfies ([H.sub.p]) for some p but [h.sub.F] = > 1.
Let F be a real abelian field satisfying ([H.sub.p]), and N = F([square root of (-p)]), K = F([[zeta].sub.p]).
Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent.
In particular, all the abelian fields F satisfying ([H.sub.3]) are determined in Carter  and the author  when [F : Q] = 2, and by Yoshimura  when [F : Q] > 2.
The main purpose of this note is to deal with real abelian fields satisfying ([H.sub.p]) for those odd prime numbers p with h(Q([square root of (-p)])) = 1, where h(Q([square root of (-p)])) is the class number of Q([square root of (-p)]).
From Proposition 1 and [5, Theorem 1.1] mentioned above, we obtain the following assertion using some computational results on abelian fields.
The reason to establish such differences is that the electron is usually described through Quantum Electrodynamics (QED) , an abelian field theory.
The authors  studied the mechanism of non-Abelian color confinement in SU(2) lattice gauge theory, in terms of the Abelian fields and monopoles extracted from the non-Abelian link variables without adopting gauge fixing.
Real abelian fields satisfying the Hilbert-Speiser condition for some small primes p ...
Frohlich, On the absolute Galois group of abelian fields, J London Math.

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