<s>[INST] Finite field, how to satisfy equation? Suppose we have $\mathbb{F}_{2^5}$ defined by polynomial $x^5+x^2+1$, and (this is homework exercise, which I kinda solved) it is required to find suitable elements $b$, so that it satisfies equation $b^2+b=x^3+1$
In this field, only 32 items and it is not a big deal to reiterate them all, here is cloud.sage script which does that.
Suitable are $b=x^7=x^4 + x^2$ with $b^2=x^{14}=x^4 + x^3 + x^2 + 1$
And $b=x^{22}=x^4 + x^2 + 1$ with $b^2=x^{44}=x^4 + x^3 + x^2$
(It is obvious that in both cases $b+b^2$ gives $x^3+1$)
It is dumb computational iteration, however. Is there any neat ways to get to this result without going through all $32$ elements?
 [/INST] Write $b = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4$, $a_i \in \{0,1\}$.  Then compute
$$b^2 + b =  (a_2 + a_4) x^4 + a_4 x^3 + (a_1 + a_2 + a_4) x^2 + (a_1 + a_3) x + a_4$$
so if this is to be $x^3 + 1$, we must have
$$ \eqalign{ a_2 + a_4 &= 0\cr
             a_4 &= 1\cr
             a_1 + a_2 + a_4 &= 0\cr
             a_1 + a_3 &= 0\cr
             a_4 &= 1\cr}$$ 
(all mod 2).  It's not hard to find the solution: $a_1 = 0$, $a_2 =1$, $a_3 = 0$, $a_4 = 1$, $a_0$ arbitrary.  So $b = x^2 + x^4$ or $1 + x^2 + x^4$. 
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