1

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.

Myshkin
  • 35,974
  • 27
  • 154
  • 332
  • 1
    There are infinitely many irreducible polynomials over $\mathbb F_2$. You need to edit your question to ask about some finite subset. – Dilip Sarwate Aug 09 '13 at 03:26

2 Answers2

3

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?

2

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 $\mathbb F_2[x]$.)

Community
  • 1
Dilip Sarwate
  • 25,197
  • 1
    In fact you don't even have to directly count how many are reducible -- you can count how many ways there are to multiply two linear polynomials together! –  Aug 09 '13 at 04:00