Multiplicative inverses for elements in field How to compute multiplicative inverses for elements in any simple (not extended) finite field? I mean an algorithm which can be implemented in software.

If 'simple' means a prime field $\mathbf{Z}/p\mathbf{Z}$ to you, then, given an integer $x$ coprime to $p$, you simply need to find an integer $y$ such that $xy\equiv1\pmod{p}.$
Look up the paragraph of multiplicative inverses in the wikipedia page on Euclidean algorithm
