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The distribution of integral and prime-integral values of systems of full-norm polynomials and affine-decomposable polynomials

Published online by Cambridge University Press:  26 February 2010

R. W. K. Odoni
R. W. K. Odoni
Affiliation:
Department of Mathematics, University of Exeter, U.K.
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Let K be any finite (possibly trivial) extension of ℚ, the field of rational numbers. Let denote the ring of integers of K, and let M ⊆ be a full module in K thus a free ℤ-module of rank [K : ℚ] contained in ; ℤ denoting the ring of rational integers. Regarding as an abelian group, the index (: M) is finite. Suppose that m1, …, mk is a ℤ-basis for M and let a ∊ Then the polynomial

(the xi; being indeterminates) will be called a full-norm polynomial; here NK/ℚ denotes the norm mapping from K to ℚ. Apart from constant factors, such a polynomial f(x) is necessarily irreducible in ℤ[x].

Type
Research Article
Information
Mathematika , Volume 26 , Issue 1 , June 1979 , pp. 80 - 87
Copyright
Copyright © University College London 1979

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