I'm having some problems following the proof for the minimum distance of irreducible goppa codes given in "A Summary of McEliece-Type Cryptosystems and their Security" by D. Engelbert, R. Overbeck, and A. Schmidt in J. Math. Crypt. 1 (2007).
Say we have a polynomial $\sigma_c$ that is a perfect square in the ring $\mathbb{F}_{2^m}$ and an irreducible polynomial $g$ such that $g$ divides $\sigma_c$. They argue that $g^2$ also divides $\sigma_c$. I think I understand why that is, but I don't see why the condition that $g$ is irreducible is relevant.
I've tried looking into other references, but none go into any more detail regarding this step, which leads me to believe that I'm missing something very simple.
Thank you for your attention.
Since $\sigma_c$ is a square, you can write $\sigma_c = p^2$ for some polynomial $p$. Now since $g$ divides $\sigma_c$ and is irreducible (and thus prime since a polynomial ring over a field is a unique factoriation domain), it follows that $g$ divides $p$, so $p = g p'$ for some polynomial $p'$. So now you have $\sigma_c = p^2 = g^2 p'^2$ which shows that $g^2$ divides $\sigma_c$.