A function in the integers module $p$ is polynomial.

Question

Let $p$ a prime number and $\mathbb{F}_p$ the field of integers module $p$. Show that if $f:\mathbb{F}_p\to \mathbb{F}_p$ is a function then $f$ is polynomial.

Answer 1

Let $t_1,...,t_p$ the elements of $F_p$, write $P(X)= \sum_{i=1}^{i=p}f(t_i){{\prod_{j\neq i}(X-t_j)}\over{\prod_{j\neq i}(t_i-t_j)}}$

Answer 2

Here’s a nonconstructive proof: The number of set-theoretic maps of $\Bbb F_p$ into itself is $p^p$, as is the number of $\Bbb F_p$-polynomials of degree $<p$. The map from (polynomials) to (self-functions) is a homomorphism, when you add two functions pointwise, and has trivial kernel, since the only polynomial of degree $<p$ that’s everywhere zero is the zero-polynomial. Thus the map is one-to-one, the sets have the same cardinality, so it’s onto.