I'm trying to decompose $\mathbb F_{9}[y]/(1+y+y^{2}+y^{3})$ into a direct sum of finite fields, but i am not sure if my approach is correct or good enough... Anyway I am not sure how to finish this problem. Can anyone help me, please? Or give me an idea?
So, first of all I've noticed that $1+y+y^{2}+y^{3}=(1+y)(1+y^{2})$. What I also know is that $\mathbb F_{9}$ is isomorphic to the field $\mathbb F_{3}[y]/(1+y^{2})$, since $1+y^2$ has no roots over $\mathbb F_{3}$. From here I am not sure how to move on...
Can anyone help me? Thank you a lot!
Write ${\mathbb F}_9 = {\mathbb F}_3[\alpha]$ with $1 + \alpha^2 = 0$, as you already indicated. Now note that over ${\mathbb F}_9$, $1 + y^2$ has $\alpha$ and $-\alpha$ as roots, so $1 + y + y^2 + y^3 = (1 + y)(1 + y^2) = (1 + y)(y - \alpha)(y + \alpha)$ over ${\mathbb F}_9$.
Using this, ${\mathbb F}_9[y]/(1 + y + y^2 + y^3)$ $\cong$ ${\mathbb F}_9[y]/(1 + y) \times {\mathbb F}_9[y]/(y - \alpha) \times {\mathbb F}_9[y]/(y + \alpha)$ $\cong$ ${\mathbb F}_9 \times {\mathbb F}_9 \times {\mathbb F}_9$; the first isomorphism here holding by the Chinese Remainder Theorem.