Inequalities for ideal bases in algebraic number fields

In a paper of nearly thirty years ago (Mahler 1937) I first studied approximation properties of algebraic number fields relative to their full system of inequivalent valuations. I now return to these questions with a slightly improved method and establish a number of existence theorems for such fields. The main result of this paper (Theorem 1) states that every ideal has a basis such that all the valuations of all the basis elements lie below Limits which can be given explicitly in terms of field constants and arbitrary parameters. Both this theorem and some of the consequences derived from it seem to be new; at least I have not found them in the recent treatments of algebraic number fields by E. Artin (1959), H. Hasse (1963), S. Lang (1964), or O. T. O'Meara (1963). The paper of 1937 depended on Minkowski's theorem on the successive minima of convex bodies (see e.g. Cassels 1959). The present paper, on the other hand, is based on a classical inequality from the reduction theory of quadratic forms, or alternatively, on a basis theorem in the geometry of numbers which was not yet known in 1937. The new approach is more powerful and enables one to construct ideal bases rather than just a system of independent elements of the ideal. I collect in § 1 the tools from the reduction theory of quadratic forms and from the geometry of numbers which are used in this paper. The next sections similarly contain the facts from valuation theory and ideal theory which are needed. In a further paper I hope to treat algebraic function fields of one variable in a similar manner.