<s>[INST] Finding the splitting field of the polynomial $x^5+2x^4+5x^2+x+4$ over $F_{11}$. I am working on the following problem:
Find the splitting field of the polynomial $f(x)=x^5+2x^4+5x^2+x+4$ over $F_{11}$.
What I have done:
So far I found that $-2$ is the only root of $f(x)$ in $F_{11}$. I have also factored $f(x)$ as $f(x)=(x+2)(x^2+5x+1)(x^2-5x+2)$ over $F_{11}$. 
Apparently the splitting field of $f(x)$ over $F_{11}$ is $F_{11^4}$. Can someone help me understand how one can conclude that.
 [/INST] The polynomial $x^2+5x+1$ has no roots in $F_{11}$: indeed
$$
x^2+5x+1=x^2-6x+9-8=(x-3)^2-8
$$
and $8$ is not a square modulo $11$.
Similarly, $x^2-5x+2=x^2+6x+9-7=(x+3)^2-7$ and $7$ is not a square modulo $11$.
Adding a root of $x^2+5x+1$ is the same as adding a square root of $2$. The elements are of the form $a+b\sqrt{2}$, and we must look whether $7$ is a square here: try and solve
$$
(a+b\sqrt{2})^2=7
$$
and you'll have your answer.
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