How to find the inverse of elements in a field?

Question

Normally I use the naive method: $$a^{-1} = a \cdot b \bmod p \equiv 1,$$ where b is the inverse of a.

Else I love to use Fermat's little theorem: $$a^{p − 1} \equiv 1 \bmod p.$$ By multiplying both sides with $a^{-1}$ you get, that the inverse is $a^{p-2}$.

But let us say I have a field of the size 8 $(F_8)$ and I shall find the inverse of $t+2$, how do I do?

Answer 1

In $\mathbb F_8$, we have $2=0$.

Assuming that $t+2\ne0$, we have $(t+2)^{-1} = t^{-1} = t^6$, because $\mathbb F_8^\times$ has order $7$.