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Ramsey’s theorem and Waring’s problem for algebras over fields. (English) Zbl 0817.12002
Goss, David (ed.) et al., The arithmetic of function fields. Proceedings of the workshop at the Ohio State University, June 17-26, 1991, Columbus, Ohio (USA). Berlin: Walter de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 2, 435-441 (1992).
Summary: It is shown that for each integer $k\geq 1$ there is an integer $R(k)$ such that every element of every algebra $A$ over every finite field $F= kF$ is a sum of difference of $k$ $k$-th powers as well as the sum of $3k/2$ $k$-th powers provided that $\text{card} (F)\geq R(k)$. Moreover, the statement of sums or differences holds also for any infinite field $F= kF$ such that the subgroup of the $k$-th powers in the multiplicative group $F\sp*$ is of finite index. Another result is that in the polynomial ring $F[ T]$ is one variable $T$ over any field $F$, every sum of cubes is the sum of 4 cubes. This result is new in the case when $\text{card} (F) =2$. For the entire collection see [Zbl 0771.00031].

##### MSC:
 12E99 General field theory 11P05 Waring’s problem and variants 11T55 Arithmetic theory of polynomial rings over finite fields 05D10 Ramsey theory