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- Title
ON A THEOREM OF DEDEKIND.
- Authors
KHANDUJA, SUDESH K.; KUMAR, MUNISH
- Abstract
Let K = ℚ(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ℚ of rational numbers. For a rational prime p, let $\bar{f}(x) = \bar{g}_{1}(x)^{e_{1}} \cdots \bar{g}_{r}(x)^{e_{r}}$ be the factorization of the polynomial $\bar{f}(x)$ obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct monic irreducible polynomials over ℤ/pℤ. Dedekind proved that if p does not divide [AK : ℤ[θ]], then the factorization of pAK as a product of powers of distinct prime ideals is given by $pA_{K} = \mathfrak{p}_{1}^{e_{1}}\cdots\mathfrak{p}_{r}^{e_{r}}$, with 픭i = pAK + gi(θ)AK, and residual degree $f(\mathfrak{p}_i/p) = {\rm deg}\, \bar{g}_{i}(x)$. In this paper, we prove that if the factorization of a rational prime p in AK satisfies the above-mentioned three properties, then p does not divide [AK:ℤ[θ]]. Indeed the analogue of the converse is proved for general Dedekind domains. The method of proof leads to a generalization of one more result of Dedekind which characterizes all rational primes p dividing the index of K.
- Subjects
DEDEKIND sums; RINGS of integers; RATIONAL numbers; POLYNOMIALS; FACTORIZATION
- Publication
International Journal of Number Theory, 2008, Vol 4, Issue 6, p1019
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042108001833