Waring's problem in algebraic number fields

Extract

Let K be a finite algebraic number field, of degree R. Then those integers of K which may be expressed as a sum of dth powers generate a subring J_{K, d} of the integers of K (J_{K, d} need not be an ideal of K, as the simplest example K = Q(i), d = 2 shows. J_{K, d} is an order, it in fact contains all integer multiples of d!; it also contains all rational integers). Siegel(12) showed that every sufficiently large totally positive integer of J_{K, d} is the sum of at most (2^d−1 + R) Rd totally positive dth powers; and he conjectured that the number of dth powers necessary should be independent of the field K—for instance, he had proved(11) that five squares are enough for every K. In this paper, we will show that, as far as the analytic part of the argument is concerned, Siegel's conjecture is correct. I have not been able to deal properly with the problem of proving that the singular series is positive; but since Siegel wrote, a good deal of extra information about singular series has been obtained, in particular by Stemmler(14) and Gray(4). The most spectacular consequence of all this is that if p is prime, then every large enough totally positive integer of J_{K, p} is a sum of (2^p + 1) totally positive pth. powers.