I've been looking at these for over an hour and I don't understand how to do them. Any hints would be greatly appreciated.
Let $p(x) = x^3 + x + 1$ and $F = Z_3[x]/\langle p(x)\rangle$. Factor $p(x)$ in $F(x)$. Does it factor into linear terms?
Here I am just supposed to factor it out only if it has a root in F(x). True?
List a complete set of representatives of $F[x]/\langle p(x)\rangle$ where $F = Z_2$ and $p(x) = x^4 + x^2 + 1$.
Any quick way to do this? Am I supposed to write down the multiplication table?
Find the multiplicative inverse of $x^2 + 1$ in $Q[x]/\langle x^4 - 2\rangle$.
Here I let $x^4 \cong 2 \mod p(x)$ and then $(x^2 + 1)y \cong 1 \mod x^4 + 2$. That gets me nowhere. However $yx^2 + y - 1 \cong x^4 - 2$. Do I have to find linear factors now?