Norms of Ideals in Direct Sums of Number Fields and Applications to the Circulants Problem of Olga Taussky-Todd

This thesis, 'Norms of Ideals in direct sums of number fields and applications to the circulants problem of Olga Taussky-Todd,' presents wide-ranging material in the Mathematical areas of Algebraic and Analytic Number Theory. The work, which is substantially original, is set out in three chapters which are supported by appendices. As the title suggests, the main aim is to tackle a problem which was originally posed by Olga Taussky-Todd who asked what values can be taken by the determinant of a certain type of n X n matrix with integer entries - the circulant (see [15]). Hitherto fragmentary algebraic results have been proved by M.Newman, using matrix manipulation ([5],[6]). However, for a given circulant, he gave no indication as to what proportion of integers are values. The thesis solves this problem by utilising a well-known relationship between determinants of matrix transformations and "absolute" norms of fractional ideals in a direct sum of number fields. By working appropriately in the latter structure, asymptotic methods are made available to complete the solution. A sketch of the mathematical strategy is given in the preface. The overall approach is to start at the level of great generality in Chapter 1 where, by slight modification, there is a generalisation of some extensive results published by R.W.K.Odoni in recent mathematical journals (see e.g. [12]). Subsequently there is successive specialisation down to the case of the circulant. In Chapter 2, by using standard techniques of group characters and the arithmetic of cyclotomic fields there are proved a few new results for abelian group determinants. In the final chapter there are given new elementary proofs of results for particular circulants, first presented by Newman in [5,6]. Then the methodology of the first chapter is reprised to establish the most important original result of this thesis - that "almost all" integers with appropriate 'critical' exponents are values of a given circulant