1

Can we find all the irreducible polynomials of $F_2[x]$ of a degree $n$?

Is the number of irreducible polynomial of $F_2[x]$ Infinite?

I was to find if there is any degree $n$ such that there is no irreducible polynomial of $F_2[x]$ of that degree.

Can anyone help me by giving some hints? I think my three doubts are related to each other.

Tianlalu
  • 5,177
cmi
  • 3,371
  • I've voted to close. This is not a strict duplicate, but a quick search would yield https://math.stackexchange.com/questions/32197/find-all-irreducible-monic-polynomials-in-mathbbz-2x-with-degree-equal?rq=1 which answers basically all of your questions – Andres Mejia Dec 14 '18 at 06:01
  • Please read my question carefully.. Is your link answering my question or giving any hint to solve it? Your link asked to find out all those polynomial whose degrees are less than or equal to five.@AndresMejia – cmi Dec 14 '18 at 06:06
  • If $f \in \mathbb{F}_2[x]$ is irreducible then $\mathbb{F}_2[x]/(f)$ is a field with $q = 2^{\deg(f)}$ elements. Its multiplicative group is of order $q-1$ so all its elements are roots of $x^q-x$. But $x^q-x$ has at most $q$ roots. Thus $\mathbb{F}_2[x]/(f)$ is exactly the splitting field of $x^q-x$ and for each $n$ there must be a splitting field of $x^{2^n}-x$ and an irreducible polynomial of degree $n$. Counting the number of irreducible polynomials of a given degree is obtained with the inclusion exclusion principle. https://www.maa.org/sites/default/files/Chebolu11739.pdf – reuns Dec 14 '18 at 06:20
  • 1
    I am writing the comment here rather than on the answer to not bother the answerer with more pings. But if you are looking for help, it will probably be a good idea to not spam people with requests for clarification every 15 minutes. – Tobias Kildetoft Dec 14 '18 at 07:17

1 Answers1

-1

The first question is hard. To answer the second and third question:

The Necklace polynomial $M(2,n)$ simultaneously represents:

  1. The number of aperiodic necklaces which can be made by arranging $n$ colored beads, from $2$ colors; and
  2. The number of monic irreducible polynomials of degree $n$ over a finite field with $2$ elements.

It is clear that this is strictly positive for all $n>0$ (e.g. one white bead and the rest black beads), hence there is at least one monic irreducible polynomial of each positive degree (over $F_2$). In particular, there are infinitely many monic irreducible polynomials (over $F_2$).

vadim123
  • 82,796
  • I did not get how can you get at least one monic polynomial for every degree $n$?@vadim123 – cmi Dec 14 '18 at 06:38
  • Can you please explain ?I am really having hard time to get your answer.@vadim123 – cmi Dec 14 '18 at 06:43
  • Can you please elaborate it? I am really not being able to understand your answer.@vadim123 – cmi Dec 14 '18 at 07:03
  • There is at least one necklace with $n$ beads, that is aperiodic. Aperiodic means you can't rotate the necklace less than 360 degrees to get the same necklace. I gave the specific necklace. Since there is at least one such necklace with $n$ beads, then there is at least one monic irreducible polynomial of degree $n$. – vadim123 Dec 14 '18 at 15:29
  • I did not get anything. How necklace comes in to the picture?@vadim123 – cmi Dec 14 '18 at 15:36
  • Click on the link in my answer. – vadim123 Dec 14 '18 at 15:56