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

## 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 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 [A_{K} : Z[θ ]]; then the factorization of pAK as a product of powers of distinct prime ideals is given by pA_{K} = p^{e}1 _{1} ···p^{er} _{r} ; with pi = pA_{K} + g_{i}(θ )A_{K}; and residual degree ƒ(p_{i}=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 [A_{K} : 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 |
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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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