Good evening! I'm studing something about finite fields and I didn't find anything that can answer my question.
I have a finite field $\mathbf{F}_q$ and the equation to solve is $x^n=a$. Which are the condition to resolve this equation? For which values of $q, n, a$ there exist a solution?
I will be very happy if anyone can answer this question, possibly with a reference of a book (or an article) that explai this with more details.
Thank in advance to everyone who can answer this one, I really need to understand this.
The case $a=0$ is easy to solve, let us look at $a \neq 0$.
The field $\mathbf{F}_q$ is cyclic. Let $b$ be a generator for the multiplicative group $\mathbf{F}_q^x$. Remember that the order of $b$ is $q-1$.
Your particular $a$ can be written as $b^k$ for some $k$. Writing $x=b^y$ the equation becomes $$ ny \equiv k \pmod{q-1} \,. $$
In particular, if $gcd(n, q-1)=1$ there is always unique solution. If $gcd(n, q-1)=d>1$, then there are solutions if and only if $d|k$, and this is equivalent to $$a^{\frac{q-1}{d}} =1$$