TY - JOUR

T1 - Algebraic gamma monomials and double coverings of cyclotomic fields

AU - Das, Pinaki

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2000

Y1 - 2000

N2 - ABSTRACT. We investigate the properties of algebraic gamma monomialsthat is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex SK, to compute H*(±,U), where U is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension K/F, we define a double covering of K/F to be an extension K/K of degree ≤ 2, such that K/F is Galois. We demonstrate that each class a ε H2(±,U) gives rise to a double covering of Q(χ∞)/Q, by Q(χ∞, √sin a)/Q(χ∞). When a lifts a canonical basis element indexed by two odd primes, we show that this double covering can be nonabelian. However, if a represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of Q(χ∞,√sina)/Q is abelian and hence √sina ε Q(χ∞). The Vsina may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

AB - ABSTRACT. We investigate the properties of algebraic gamma monomialsthat is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex SK, to compute H*(±,U), where U is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension K/F, we define a double covering of K/F to be an extension K/K of degree ≤ 2, such that K/F is Galois. We demonstrate that each class a ε H2(±,U) gives rise to a double covering of Q(χ∞)/Q, by Q(χ∞, √sin a)/Q(χ∞). When a lifts a canonical basis element indexed by two odd primes, we show that this double covering can be nonabelian. However, if a represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of Q(χ∞,√sina)/Q is abelian and hence √sina ε Q(χ∞). The Vsina may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

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U2 - 10.1090/s0002-9947-00-02417-x

DO - 10.1090/s0002-9947-00-02417-x

M3 - Article

AN - SCOPUS:23044517634

VL - 352

SP - 3557

EP - 3594

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -