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I’m studing for an exam and I am stuck on the following practice problem.

Consider the the ring $R=\mathbb{F}_{p}[x]$. How many irreducible polynomials of degree 4 exist in $R$?

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Irreducible quartic polynomials (assumed monic) in $R$ are exactly the minimal polynomials of those elements of the (up to isomorphism) unique finite field $L=\mathbb{F}_{p^4}$ of $p^4$ elements that don’t belong to a proper subfield.

From the basic facts about containment of one finite field in another we immediately see that $K=\mathbb{F}_{p^2}$ is (isomorphic to) the unique maximal subfield of $L$. Therefore there are exactly $p^4-p^2$ elements in $L$ with quartic minimal polynomials. As the said minimal polynomials are separable (owing to the fact that the base field is perfect), each and every one of them is the minimal polynomial of exactly four conjugate elements. Therefore there are

$$

N_p=\frac{p^4-p^2}4

$$

irreducible monic quartic polynomials with coefficient in $\mathbb{F}_p$.

Note also that uniqueness of $L$ means that any irreducible quartic is the minimal polynomial of some element of $L$. This is because any such quartic has a root in some quartic extension of $\mathbb{F}_p$, but $L$ is the only one.

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