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Once you have the assumption that the splitting field is of the form $\mathbb{F}_{5^n}$ for some $n$ the question is quite simple, but how does one actually get to that point? Why is it true that a splitting field of $X^7-1$ over $\mathbb{F}_5$ is of such form? Is this true in a more general case for splitting fields over finite fields?

In related posts like here, In the case of $(x^3+x-1)$ over $\mathbb{F}_3$ its claimed that since its irreducible, it follows that the splitting field is $\mathbb{F}_{3^4}$. So, does this have to do with irreducibility? Separability?

user1591021
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1 Answers1

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Any finite field extension $L$ of $K$ is a $K$-vector space. So additively, we have $L\simeq K^n$ for $n=[L:K]$. When $K$ is a finite field, $|L|=|K|^n$ is fixed, and it's a general theorem that the structure of a finite field is completely determined by its size.

In the case of finite fields, all extensions are Galois, i.e. normal and separable, so we never have to worry about separability, and the splitting field of an irreducible polynomial is just to add a single root. That is, if $f(x)$ is irreducible over finite field $K$, then its splitting field is just $K[x]/(f(x))$ which has size $|K|^{\deg f}$ (this only works when $f$ is irreducible, for otherwise $K[x]/(f(x))$ is not an integral domain). Especially, since $x^3-x+1$ is irreducible over $\mathbb F_3$ (check it has no root in $\mathbb F_3$), its splitting field is $\mathbb F_{3^3}$, not $\mathbb F_{3^4}$.

Just a user
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  • Sorry, I removed my comment because I was not sure, so I will restate it if that's ok. I just want to make sure I am understanding what you are saying. So of course for your setup (always) $L$ is a $K$ vector space, so if $K = \mathbb{F}p$ for some $p$ prime, then it must be that the cardinality of $L$ = $p^n$, and since its a field we know $L \cong \mathbb{F}{p^n}$ due to the uniqueness of finite fields? Also, thank you for the last paragraph. I hadn't thought of this. – user1591021 Mar 11 '25 at 22:22
  • You got it right. Mostly these are just facts which eventually one gets used to, not to be meant in logical order. From time to time, you'll have to take a step back, have a look at the techniques used in proofs (not necessarily the full proofs) to understand "why", such as to use the an extension is a vector space over the base, and to use $K[x]/(f(x))$ to construct field extensions, as in Kronecker's theorem. – Just a user Mar 11 '25 at 22:35