I know there are finite fields like $\mathbb F_2$, $\mathbb F_4$ or the $\mathbb Z/n\mathbb Z$ for prime $n$ with modulo operations. For other special $n$, I've seen fields $\mathbb F_n$ with $n$ elements being constructed. And of course there are the usual infinite fields (take $\mathbb Q$ and so on).
So I wonder: Let $M$ be an arbitrary set that contains at least two elements. Can you always find operations $+ : M\times M \to M$ and $\cdot : M \times M \to M$, such that $(M, +, \cdot)$ is a field?