This question raised another question in my mind. In finite fields $F$, every function from $F$ to itself is a polynomial. What about infinite fields of finite (i.e. non-zero) characteristic? Are there non-polynomial functions there?
---- Naively
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3$\begingroup$ Sure, non-polynomial is any function $\ne 0$ with infinitely many roots, e.g. delta functions, step functions, etc. $\endgroup$– Bill DubuqueMar 7, 2012 at 4:38
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3 Answers
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If a field is infinite, of whatever characteristic, the function that takes value $1$ on $0$ and $0$ on $x\neq0$ is not polynomial.
answered Jan 26, 2012 at 21:24
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If a field $F$ has infinite cardinality $\kappa$, then there are $\kappa^\kappa=2^\kappa$ functions from $F$ to $F$, but only $\kappa$ polynomials over $F$. Since $2^\kappa > \kappa$, there must be non-polynomial functions.
answered Jan 26, 2012 at 21:37
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$\begingroup$ I don't know why I didn't think of this one. If your whole answer had been just one word, "cardinality", I'd have had it in an instant. Haste makes waste. $\endgroup$ Jan 26, 2012 at 21:40
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The full story about polynomials versus polynomial functions over an integral domain is told in $\S 14.1$ of my undergraduate number theory notes. (In particular you will find both Andrea's and Chris's answers in there.)
answered May 22, 2012 at 18:04