Splitting field of cyclotomic polynomials over $\mathbb{F}_2$. Let $\Phi_5$ be the 5th cyclotomic polynomial and $\Phi_7$ the 7th. These polynomials are defined like this:
$$
\Phi_n(X) = \prod_{\zeta\in\mathbb{C}^\ast:\ \text{order}(\zeta)=n} (X-\zeta)\qquad\in\mathbb{Z}[X]
$$
 I want to calculate the splitting field of $\Phi_5$ and the splitting field of $\Phi_7$ over $\mathbb{F}_2$. In $\mathbb{F}_2[X]$ we have
$$
\Phi_5(X) = X^4 + X^3 + X^2+X+1
$$
and
$$
\Phi_7(X) = (X^3+X+1)(X^3+X^2+1)
$$
My question is: what are the splitting fields of the polynomials? I already know it should be of the form $\mathbb{F}_{2^k}$ for some $k\in\mathbb N$. Also the degree of every irreducible factor of a cyclotomic polynomial in $\mathbb{F}_q[X]$ is equal to the order of $q\in(\mathbb{Z}/n\ \mathbb{Z})^\ast$, assuming $(q,n)=1$.
 A: Since we want the degree of an irreducible factor to be equal to one, we want
$$
\text{order} (2^k) =1
$$
in $(\mathbb{Z} / 5\mathbb{Z})^\ast$. The only element with this order is 1. Therefore we search the smallest $k$ such that $2^k\equiv 1\mod 5$. A bit puzzzling gives us
$$
2^1=2\\
2^2=4\\
2^3=8=3\\
2^4=16=1.
$$
Therefore the splitting field of $\Phi_5$ should be $\mathbb{F}_{2^4}$.
Is this correct?
A: Hint :


*

*If $f(X)$ is irreducible in $F[X]$ then $F[X]/(f(X))$ is a field.

*any polynomial $f(X)$ has root in $F[X]/(f(X))$ (which need not be a field in general)

*What is  the cardinality of $\mathbb{F}_2[X]/(X^4 + X^3 + X^2+X+1)$.

*How many finite fields of cardinality $n$ can you list out for a given $n$.

*Splitting field of $f(X)g(X)$ is contains splitting field of $f(X)$ 

*Splitting field of $(X^3+X+1)(X^3+X^2+1)$ is contains splitting field of $(X^3+X^2+1)$

*As $(X^3+X^2+1)$ is irreducible in $\mathbb{F}_2[X]$ its splitting field would be (???)

*What is splitting field of $(X^3+X+1)$.

*Do you see some relation between splitting field of $(X^3+X+1)$ and of $(X^3+X^2+1)$.


Can you now conclude??
