Polynomials in Finite Field

Question

Given $\Bbb F_{32}=\Bbb F_2[X]/(X^5+X^2+1)$ where the polynomial is irreducible over $\Bbb F_2$, how would I compute $(a^4+a^2)*(a^3+a+1)$ given $a=[X]$ is the congruence class of [X]? Multiplying the polynomials acquires $a^7+2 a^5+a^4+a^3+a^2$, but how do I reduce that to fit the given conditions?

Answer 1

Use the Polynomial Division Algorithm (with Remainder) to write it as $\, q(a) (a^5\!+a^2\!+1)+ r(a),\,$ where $\,\deg r < 5.\,$ Or, equivalently, use the equation $\, \color{#c00}{a^5 = a^2+1}\,$ as a rewrite rule to reduce all exponents on $\,a\,$ till they are $\, < 5,\,$ e.g. $\,a^{k+5} = a^k \color{#c00}{a^5} = a^k(\color{#c00}{a^2+1}).\,$