In this paper, we formulate an analogue of Waring's problem for an algebraic group G. At the field level we consider a morphism of varieties f : A 1 → G and ask whether every element of G(K) is the product of a bounded number of elements f (A 1 (K)) = f (K). We give an affirmative answer when G is unipotent and K is a characteristic zero field which is not formally real.The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of G(O) can be written as a product of a bounded number of elements of f (O). We prove this is the case when G is unipotent and O is the ring of integers of a totally imaginary number field.
Introduction 1 2. The splitting field of the polynomial C a (x) − m 3 3. An analogue of the Grunwald-Wang theorem for certain Galois extensions K/k 8 References 11
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