We consider the polynomial $p(X) := X^2 + X + 2$ in the ring $\mathbb{F}_3[X]$.
(a) Show that $E := \mathbb{F}_3[X]/ p(X)$ is a finite field extension of $\mathbb{F}_3$.
(b) Determine a basis of $E$ as a vector space over $\mathbb{F}_3$. For all elements of $E$, give their multiplicative inverses.
(c) Using the polynomial $q(X) := X^2 + 2X + 2$, we continue to consider the finite field extension $F := \mathbb{F}_3[X]/ q(X)$ of $\mathbb{F}_3$. Determine a field isomorphism $ϕ: E → F$ that leaves $\mathbb{F}_3$ fixed element-wise.
For a) my idea was to check the field extension $\mathbb{F}_3(a)/\mathbb{F}_3$ where $a$ is a root of the polynomial $p(X)$. Since $\mathbb{F}_3(a)/\mathbb{F}_3$ is isomorphic to $ \mathbb{F}_3[X]/ p(X)$ if we can show that is a finite field extension we have the answer . I can prove that is a field extension since $p(X)$ is irreducible over $\mathbb{F}_3$ and since $p(X)$ is an irreducible polynomial of degree $2$ $\mathbb{F}_3(a)/\mathbb{F}_3$ is a finite field extension and therefore (because of $\mathbb{F}_3(a)/\mathbb{F}_3 \cong E$) we have that $E$ is a finite field extension. So far everything right?
For b) I dont know how to proceed. I know that every element of E can be rapresented as $e=a+bp(X)$ where $a,b \in \mathbb{F}_3[X]$, but from here I dont know how to proceed. Can someone help me?
For c) I have really no idea.
Can someone help me?
This should be a comment, but it is too long.
Denote $(X^2+X+2)=I$. So we have the diagram: (Note that $\mathbb{F}_3[X]/I$ is not an extension of $\mathbb{F}_3$ in the set-theoretical sense. $\mathbb{F}_3[X]/I$ is an extension of $(\mathbb{F}_3+I)/I$ in the set-theoretical sense, and we have the obvious isomorphism $(\mathbb{F}_3+I)/I\cong \mathbb{F}_3$ as indicated in the diagram.)
Since $\{1,a\}$ is a basis for $\mathbb{F}_3(a)$ as a vector space over $\mathbb{F}_3$, and they are image of $1+I$ and $x+I$ under the isomorphism $ g(x)+I\mapsto g(a)$ respectively, therefore $1+I$ and $x+I$ is a basis for $\mathbb{F}_3[X]/I$ as a vector space over $(\mathbb{F}_3+I)/I$.