Modulo equation with prime

Question

Let $p$ be prime with $p\equiv 2 \bmod 3$. Show that for any integer $a,p\nmid a$ there is an integer $x:x^3 \equiv a\bmod p$

Not sure that I found the best match for this oft asked question. Andreas' work in teaching to think about this is, of course, worthy of an upvote. — Jyrki Lahtonen, Aug 03 '14 at 11:07

score 4 · Accepted Answer · answered Aug 03 '14 at 11:03

Hint 1

Consider the multiplicative group $G$ of integers modulo $p$. This is an abelian group of order $p-1$.

Hint 2

Since $p-1 \equiv 1 \pmod{3}$, you have that $\gcd(p-1, 3) = 1$.

Hint 3

Prove that the map $f : x \mapsto x^{3}$ is a homomorphism, and that is injective, by computing its kernel.

Hint 4

Now $f$ is an injective map on the finite set $G$, thus it is also surjective.

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