# non-polynomial functions on fields of finite characteristic

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

• Sure, non-polynomial is any function $\ne 0$ with infinitely many roots, e.g. delta functions, step functions, etc. Mar 7, 2012 at 4:38

If a field is infinite, of whatever characteristic, the function that takes value $1$ on $0$ and $0$ on $x\neq0$ is not polynomial.
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.
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.)