# Number of irreducible polynomial over a field. [closed]

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

## closed as off-topic by user228113, Morgan Rodgers, Kushal Bhuyan, JMP, user1551Jul 17 '16 at 15:41

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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.

• I think this answer is not really useful for anybody that might actually ask the question. In fact it may mislead somebody into believing the significantly more complicated result is needed. (I will not add a second downvote though.) – quid Jul 17 '16 at 12:17

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.