number of quadratic Ireducible Polynomial over the field $\mathbb{F}_2$

Question

Could anyone help me : What is the way of finding the number of Ireducible quadratic Polynomial over the field $\mathbb{F}_2$?

Thank you for help.

Answer 1

For a general answer, see, for example, here.

For your particular problem, just make a list of the four quadratic polynomials in $\mathbb F_2[x]$ and figure out how many of them are reducible. Then count the rest to come up with the answer. (Hint: $x^2+1 = (x+1)^2$ in $\mathbb F_2[x]$.)

Answer 2

An irreducible quadratic polynomial over $\mathbb{F}_2$ has a conjugate pair of roots in $\mathbb{F}_4$ that are not in any proper subfield of $\mathbb{F}_4$. ($\mathbb{F}_2$ is the only proper subfield)

So how many elements are there in $\mathbb{F}_4$ that aren't in any subfield? How many conjugacy classes are there?