2

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

Please anyone help me.

quid
  • 42,135
chandan
  • 334
  • 2
  • 12

2 Answers2

6

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.

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
2

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.

Community
  • 1
Kushal Bhuyan
  • 7,110
  • 5
    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