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Bases for integer-valued polynomials in a Galois field. (English) Zbl 0923.11164
Let $$GF[q,x]$$ be the ring of polynomials over the Galois field $$GF(q)$$ of characteristic $$p$$ with $$q=p^n$$ and $$GF(q,x)$$ its quotient field. An integer-valued polynomial is a polynomial $$f(T)\in GF(q,x)[T]$$ such that $$f(M)\in GF[q,x]$$ for all $$M\in GF[q,x]$$. Denote by $$D^r$$, $$r\in \mathbb{N}$$, respectively $$D^\infty$$ the class of integer-valued polynomials which together with their derivatives up to order $$r$$, respectively of all orders, are integer-valued. Denote also by $$\Delta^r$$, $$r\in\mathbb{N}$$, respectively $$\Delta^\infty$$, the class of integer-valued polynomials which together with their differences up to order $$r$$, respectively of all orders, are integer-valued. Bases for $$\Delta^r$$ and $$\Delta^\infty$$ over $$GF[q,x]$$ are known. The author gives bases for $$D^1$$ and $$D^2$$ over $$GF[q,x]$$ and proves that for a positive integer $$r$$, we have $$\Delta^r\subset D^r$$ and $$\Delta^\infty= D^\infty$$.

##### MSC:
 11T06 Polynomials over finite fields 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 11C08 Polynomials in number theory 11T55 Arithmetic theory of polynomial rings over finite fields
##### Keywords:
bases; Galois field; integer-valued polynomial
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