Showing polynomial $p(x) = x^2 −3 \in\mathbb{F}_7[x]$ is prime in $E[x]$ and $x^3 - 2\in\mathbb{F}_7[x]$ factors into linear terms in $E$

Question

Here we define the set of equivalence classes $E[x] = \mathbb{F}_7[x]/(x^3 - 2)\mathbb{F}_7[x]$. I'm not sure if showing $p(x)$ is prime is equivalent to showing that it is a primitive root of $E^\times$? I'm not sure how to do either if that's the case.

Answer 1

(As a general rule, you should restate your question fully in the body, and not split it between title and body.)

Following you, I call $E$ the ring $\Bbb F_7[x]/\langle x^3-2\rangle$. Note that $x^3-2$ has no roots in $\Bbb F_7$, so is irreducible, so that $E$ is a field; you may think of $E$ as $\Bbb F_{7^3}=\Bbb F_{343}$.

I’m going to name $\xi$ as the image of $x$ in $E$. You might as well think of $\xi$ as a cube root of $2$. Accordingly, every element of $E$ can be written $k+m\xi+n\xi^2$ for elements $\{k,m,n\}$ of $\Bbb F_7$, in other words integers between $0$ and $6$ inclusive. Thus every equivalence class is of the form $(k+mx+nx^2)+(x^3-2)\Bbb F_7[x]$.

In the ring of polynomials over $E$, you don’t want to call the indeterminate $x$; I’ll call it $T$, so we’re interested in factoring $T^3-2$ in $E[T]$, and since one root is $\xi$, the factors are $T-\xi$ and what you get by replacing $\xi$ there by its product with (nontrivial) cube roots of unity. Fortunately these are already in $\Bbb F_7$, and represented by integers. So the complete factorization is $T^3-2=(T-\xi)(T-2\xi)(T-4\xi)$, which you should check.

I think that takes care of everything except your questions about $T^2-3\in E[T]$.

Your first question about $T^2-3$ is whether it’s prime, i.e. irreducible, in $E[T]$. Here, you are asking about square roots of $3$; since there are none in $\Bbb F_7$, the question is whether $3$ has a square root in $\Bbb F_{343}\cong E$. If not, $T^2-3$ is still irreducible (prime) in $E[T]$. Notice that $3$ is a primitive sixth root of unity in $\Bbb F_7$, so its square root will be a primitive twelfth root of unity in its extension field, whatever that may be. But the order of $\Bbb F_{343}^\times$ is $342=2\cdot3^2\cdot19$, not divisible by twelve. So $T^2-3$ is still irreducible (prime) over your larger field.

Finally, you ask whether $\xi^2-3$ has order $342$ in $E^\times$. This is quite separate from the other questions; that order is indeed $342$ (by a method I don’t want to go into), but this does seem to be totally unrelated to the fact that $T^2-3$ is prime in $E[T]$.

Good luck to you. I hope this answered your quesitons.