A Generalization of Dedekind Criterion

Abstract

Let K = ℚ(θ) be an algebraic number field with θ in the ring A _K 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 (x) = (x) ^{e ₁} …(x) ^{e _r} be the factorization of the polynomial (x) obtained by replacing each coefficient of f(x) modulo p into product of powers of distinct irreducible polynomials over ℤ/pℤ with g _i (x) monic. In 1878, Dedekind proved that if p does not divide [A _K :ℤ[θ]], then , where ℘₁,…, ℘ _r are distinct prime ideals of A _K , ℘ _i  = pA _K  + g _i (θ)A _K with residual degree f(℘ _i /p) = deg (x). He also gave a criterion which says that p does not divide [A _K :ℤ[θ]] if and only if for each i, we have either e _i  = 1 or (x) does not divide where . The analog of the above result regarding the factorization in A _K′ of any prime ideal 𝔭 of A _K is in fact known for relative extensions K′/K of algebraic number fields with the condition “p ≠ | [A _K :ℤ[θ]]” replaced by the assumption “every element of A _K′ is congruent modulo 𝔭 to an element of A _K [θ]^(†)”. In this article, our aim is to give a criterion like the one given by Dedekind which provides a necessary and sufficient condition for assumption (†) to be satisfied.

Key Words:

• Factorization of prime ideals
• Ramification and extension theory

2000 Mathematics Subject Classification:

• 11S15
• 11Y05

ACKNOWLEDGMENTS

The authors are highly thankful to Dr. Peter Roquette Emeritus Professor Universität Heidelberg for having given us the idea for the proof of the main theorem. The financial support by University Grants Commission, New Delhi and National Board for Higher Mathematics, Mumbai is gratefully acknowledged.

Notes

Communicated by U. Otterbeck.