Determining orders of elements in quotient ring with SAGE

Question

I wish to determine whether certain elements of the quotient ring $Q:=\mathbb{F}_{2}[x]/\langle (x^{5}-1)^2\rangle$ are units and, in the case that they are indeed units, I would also like to compute their multiplicative orders. How can one go about doing this?

In case you consider that my previous questions is a wee bit too broad, I have a related question which is more specific: if $p(x) \in \mathbb{F}_{2}[x]$ and $k \in \mathbb{N}$, how does one even determine a representative of the class (in the ring $Q$) to which $(p(x))^{k}$ belongs? The naïve approach does not seem to work here: in my viewpoint, the code

F = GF(2)

R. = PolynomialRing(F)

S. = R.quo((x^5-1)^2)

b=x^5-1

b^2

SHOULD output [0] because the image of $(x^{5}-1)^2$ under the natural projection $\mathbb{F}_{2}[x] \to Q$ is exactly equal to the zero element of the quotient ring $Q$, but it does not yield that (it outputs the polynomial $x^{10}+1$ instead, duh!). Do you know how it is that I am supposed to modify it in order to get the actual class in $Q$ to which $(x^{5}-1)^{2}$ belongs?

Thanks in advance for your insightful replies!

Answer 1

See also in magma

   F := GF(2);
   R<x> := PolynomialRing(F);
   I := ideal<R|(x^5-1)^2>;
   S<y> := quo<R|I>;

   for a in [0..1] do
    for b in [0..1] do
      c := a+b*y;
      if IsUnit(c) then
        [c,c^(-1)];
      end if;
     end for;
   end for;