# Class to which an element belong in a quotient ring

Fellow Sages:

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 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 quotient ring $Q$) to which $(p(x))^{k}$ belongs? The naïve approach does not seem to work here: in my viewpoint, the code

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 $x^{10}+1$, duh!). Do you know how it is that I am supposed to modify it in order to get the class in $Q$ to which the power in question belongs? Thanks in advance for your insightful replies!F = GF(2)

R.<x> = PolynomialRing(F)

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

b=x^5-1 b^2;