A norm residue map for central extensions of an algebraic number field
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Let K be a finite Galois extension of an algebraic number field k with G = Gal (K/k), and M be a Galois extension of k containing K. We denote by resp. the genus field resp. the central class field of K with respect to M/k. By definition, the field is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k. Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H-3(G, Z), and is isomorphic to H-3(G, Z) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).
1984 ◽
Vol 93
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pp. 133-148
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1957 ◽
Vol 12
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pp. 177-189
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1991 ◽
Vol 121
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pp. 161-169
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1967 ◽
Vol 29
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pp. 281-285
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1961 ◽
Vol 19
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pp. 169-187
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1987 ◽
Vol 107
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pp. 135-146
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2010 ◽
Vol 06
(06)
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pp. 1273-1291
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