Degree of splitting field over $\mathbb{F}_{2^k}$ and $\mathbb{F}_{3^k}$.

Question

I want to calculate the degree of splitting field over $\mathbb{F}_{2^k}$ and $\mathbb{F}_{3^k}$ of $f(x)=x^4+3x^3+x+1$, when $k\ge 1$.

Answer 1

I'll show how it works over $\mathbb{F}_{2^k}$.

1. Observe that $x=1$ is a root of the given polynomial. Then we can write $$ x^4+3x^3+x+1=(x+1)(x^3+1). $$ Therefore, it is enough to answer the question for $x^3+1$.

2. Observe that $x=1$ is a root of $x^3+1$. Then, we can write $$ x^3+1=(x+1)(x^2+x+1). $$ Therefore, it is enough to consider the question for $x^2+x+1$.

3. Observe that $x^2+x+1$ is irreducible over $\mathbb{F}_2$ since it is of degree $2$ and neither $0$ nor $1$ is a root.

4. Therefore, since $x^2+x+1$ is of degree $2$, in $\mathbb{F}_{2^k}$, it is either irreducible or splits completely. Since $x^2+x+1$ splits completely in $\mathbb{F}_{2^2}$, it splits in $\mathbb{F}_{2^k}$ iff $\mathbb{F}_{2^k}$ contains $\mathbb{F}_{2^2}$. In other words, exactly when $2\mid k$.

Therefore, if $k$ is even, $x^2+x+1$ splits in $\mathbb{F}_{2^k}$. If $k$ is odd, then $x^2+x+1$ splits in the degree $2$ extension $\mathbb{F}_{2^{2k}}$.