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.

In both cases one may employ the extended Euclidean algorithm to compute inverses. See here for an example. Alternatively, employ repeated squaring to compute $\rm\:a^{-1} = a^{q-2}\:$ for $\rm\:a \in \mathbb F_q^*\:,\:$ which is conveniently recalled by writing the exponent in binary Horner form. A useful reference is Knuth: TAoCP, vol 2: Seminumerical Algorithms.
