Number of irreducible polynomial over a field.

Question

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements.

Please anyone help me.

Answer 1

Note: As this result is given in this answer.

The number of irreducible monic polynomials of degree $n$ (only when $n$ is prime) over the field of characteristic $p$ is $\frac{p^n-p}{n}$. In your case $p=5$ and $n=2$

Alternately Gauss gave the following result,

The number of irreducible monic polynomials of degree $n$ over $F_q$ is given by $$N_q(n)=\frac{1}{n}\sum_{d|n}\mu(d)q^{n/d}$$ where $\mu$ is the Mobius function.

For a proof see this.

Answer 2

Hints.

1. Count the number of monic quadratic polynomials, irreducible or not.

2. Count the number of products $(x-a)(x-b)$, remembering that $a$ might equal $b$.

3. Subtract.