I know that in arbitrary division rings, one can go about finding inverses Euclidean division. But take $\mathbb Z_{11}$ as a simple example. Is there a "nice" expression which yields the inverses in general? i.e. an expression $e(n)$ such that
$$ e(1) = 1, \quad e(2) = 6, \quad e(3) = 4, \quad e(4) = 3, \quad e(5) = 9, \quad e(6) = 2, \quad e(7) = 8, \quad e(8) = 7, \quad e(9) = 5, \quad e(10) = 10, $$
all up to mod 11.
I tried polynomial interpolation, but ended up with this ugly thing:
$$-\frac{7 x^9}{2160}+\frac{77 x^8}{480}-\frac{4279 x^7}{1260}+\frac{28919 x^6}{720}-\frac{4653 x^5}{16}+\frac{1911679 x^4}{1440}-\frac{4091593 x^3}{1080}+\frac{2321143 x^2}{360}-\frac{1229503 x}{210}+2123,$$
which isn't surprising, given the points it should pass through:
Naturally this does not account for modulo 11, so probably one can get something better if polynomial interpolation can be adapted up to mod 11. Or maybe the expression isn't a polynomial at all, maybe it can include a factorial term? I'm mentioning this because I tried to play around with Wilson's theorem but this didn't yield anything immediately useful.
Yes, there is a nice and simple polynomial (over the field $\mathbb F_p$): namely, $$ x \mapsto x^{p-2}. $$ If you want to map the integers $[0,p-1]$ to themselves, combine this with the fractional part function: $$ x\mapsto p\left\{\frac{x^{p-2}}{p}\right\}. $$