Fractional exponentiation in modular arithmetic

Question

Does raising a modular expression to a fraction mean anything?

For example, $a\,\,mod \,\,N$ raised to $1/b$ where $b>0$.

Does this violate the rules of modularity?

Answer 1

It depends. For example, whenever $N$ is prime and $N \equiv 3 \pmod{4}$, then $(-1)^{\frac{1}{2}}$ doesn't make sense, since there are no elements whose square is $-1$. However, when $N$ is prime and $N \equiv 1 \pmod{4}$ then this (almost) makes perfect sense because $-1$ is a square. However, there is no clear choice of a square root. For example, when $N=5$, both $2$ and $3$ square to $-1$, but none is a "better" square root than the other. Hope that helps.