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The Hooley–Huxley contour method, for problems in number fields II: factorization and divisibility
Published online by Cambridge University Press: 26 February 2010
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Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in 
, the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define 
. Then the small region of K referred to above is
for 0<l<½ and with the notation that, for any α ∈ ℝ, we set 
β. where β is the unique real satisyfing 
.
MSC classification
Secondary:
11R47: Other analytic theory
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- Research Article
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- Copyright © University College London 2002
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