A Study of Algebraic Number Fields

Abstract

Discriminant whose notion is due to R. Dedekind, is a basic invariant associated to newlinean algebraic number and#64257;eld. Its computation is one of the most important problems newlinein algebraic number theory. For an algebraic number and#64257;eld K = Q(and#952;) with and#952; in the newlinering A K of algebraic integers of K having F (x) as its minimal polynomial over the newlineand#64257;eld Q of rational numbers, the discriminant d K of K and the discriminant of the newlinepolynomial F (x) are related by the formula newlinediscr(F ) = [A K : Z[and#952;]] 2 d K . newlineSo computation of d K is closely connected with that of the index of Z[and#952;] in A K . We newlinecharacterize those primes which divide the discriminant of F (x) but do not divide newline[A K : Z[and#952;]] when and#952; is a root of an irreducible trinomial F (x) = x n +ax m +b belonging newlineto Z[x]. Such primes p are important for explicitly determining the decomposition newlineof pA K into a product of prime ideals of A K in view of the well known Dedekind newlinetheorem. As an immediate consequence, we obtain some necessary and suand#64259;cient newlineconditions involving only a, b, m, n for {1, and#952;, · · · , and#952; nand#8722;1 } to be an integral basis of K. newlineDiscriminant is also a valuable tool to and#64257;nd an integral basis of an algebraic newlinenumber and#64257;eld K. The problem of its computation specially for pure number and#64257;elds newlinehas attracted the attention of many mathematicians. We give an explicit formula newlineand#8730; newlinefor the discriminant of squarefree degree pure number and#64257;elds Q( m a), with x m and#8722; a newlineirreducible over Z, involving only the primes dividing m and the prime powers newlineand#8730; newlinedividing a. In case K = Q( n a) is an extension of degree n of the and#64257;eld Q of newlinerational numbers.