Algebraic integers of small discriminant

(1) D(α) = D(K)(I(α))2, where I(α) = [OK : Z[α]], the index of the ring generated by α in the full ring of integers OK of K. Classically, the index of the field K is the greatest common divisor of all indices I(α) for α ∈ OK with K = Q(α). Dedekind was the first to show that the index may not be 1 by exhibiting certain cubic and quartic fields with this property. By results of Bauer and von Żyliński early this century, it is known that a rational prime p divides the index of some field of degree d if and only if p < d. Later work by Engstrom [2] investigated which powers of these primes may occur as common index divisors. In contrast to the index of the field, we focus here on the minimal index of the primitive algebraic integers of the field, which we denote by I(K). In other words, I(K) = min{I(α)}, where the minimum is over all α ∈ OK with K = Q(α). The simplest situation is of course I(K) = 1, in which case OK is said to have a power basis. This happens trivially for any quadratic field, but when the degree of K is larger than 2 one does not expect this to be the case in general. In fact, much work has been done on the problem of classifying fields of certain types which have a power basis (in some cases with complete success). We refer the reader to the papers of Gras [3 and 4] and Cougnard [1] for more on this topic. A natural question to ask, and the one which we address here, is how large I(K) can be. In view of the remarks above on the index, this is only interesting when considering fields of degree less than a given bound. Here we determine upper bounds for I(K) in terms of just the degree and discriminant of K, with sharper bounds using more properties of the field. We