A matrix criterion for normal integral bases
Donald E. Maurer · Illinois Journal of Mathematics · 1978
Let K IF be a finite Galois extension of an algebraic number field F. In certain circumstances it is known that the ring of integers (9 has a normal integral basis.The uniqueness of such a basis has been studied in [2] and [3].In this paper we give a characterization of the structure constants of an order, over an integral domain, having a normal integral basis.As an example these results are then applied to cyclic cubic extensions" we obtain an explicit charac- terization of the normal orders of such extensions in terms of their diserimin- ants; and when F is the rational field Q, we are able to characterize discriminants of tamely ramified cyclic cubic extensions, and explicitly con- struct all such fields having a given discrirninant.It is known (e.g., class-field theory) that each cyclic extension of Q is determined by a complex-valued Dirichlet character with the property that for almost all primes p, X(p)= 1 if and only if p splits completely in the extension.For quadratic extensions the character is known explicitly in terms of the diseriminant.For higher degree extensions the corresponding character is no longer determined by the dis- erimant, but there is still a very strong connection which our results make explicit.