On a theorem of Dedekind

Khanduja, Sudesh K. ; Kumar, Munish (2008) On a theorem of Dedekind International Journal of Number Theory, 4 (6). pp. 1019-1025. ISSN 1793-0421

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Official URL: http://www.worldscinet.com/ijnt/04/0406/S179304210...

Related URL: http://dx.doi.org/10.1142/S1793042108001833


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 ƒ-(x) = g-1(x)e1 ···g-r(x)er be the factorization of the polynomial ƒ-(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 pAK = pe1 1 ···per r ; with pi = pAK + gi(θ )AK; and residual degree ƒ(pi=p) = deg 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.

Item Type:Article
Source:Copyright of this article belongs to World Scientific Publishing Company.
Keywords:Factorization of Prime Ideals; Ramification and Extension Theory
ID Code:83981
Deposited On:23 Feb 2012 12:26
Last Modified:23 Feb 2012 12:26

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