# 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 $E$

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.

(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 $$E[T]$$.