Prove that there exists a $p$-root for every member of $GF(q)$ and/or existence of power in $GF(q)$

Question

I've been baffled by this question: "Let $\mathrm{GF}(q)$, where $q$ is a prime power of $p$ ($q=p^h$), be a finite Galois field, prove that for each member of the field there exists a $p$ root of it."

I think I have a solution, but there's one thing I'm unsure about, if for every $b$ in $\mathrm{GF}(q) \rightarrow b^{(ap^-1)}$ is in $\mathrm{GF}(q)$ ($a$ is a random number) ?

Answer 1

In the answer I'm going to use the following theorems:

• every Galois-field contains a primitive element
• for all $x$, $x^q = x$
• every field is closed in relation to it's multiplication

Let $a$ be our primitive element, we define $b=a ^{p^h-1}$. Then $b^p = ( a^{p^h-1} ) ^p = a^q = a$.

Hence, since $a$ is our primitive element, for each $c$ in the field

$$c=a ^ l = (b^p) ^l = (b^l) ^ p$$

So every element has a $p$-root.