Waring's problem in algebraic number fields

Source abstract

Let K be a finite algebraic number field, of degree R . Then those integers of K which may be expressed as a sum of d th 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 d th powers; and he conjectured that the number of d th 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 p th. powers.