The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$

Asked Mar 27 '14 at 15:13

Active Mar 27 '14 at 16:03

Viewed 287 times

I meet the problem in Hungerford's Algebra, page 282 as follows:

If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ .

I have no any idea on this. But I really want to know how to prove this.

Thanks!

edited Mar 27 '14 at 16:03

Martin Sleziak

asked Mar 27 '14 at 15:13

Khako

2

These question seems to be similar (the same?): http://math.stackexchange.com/questions/663398/irreducibility-of-x2nx1-over-mathbbz-2 and http://math.stackexchange.com/questions/508920/on-irreducible-factors-of-x2nx1-in-mathbb-z-2x – Martin Sleziak Mar 27 '14 at 16:03
1

Voting to close as a duplicate. The answers to questions in Martin's links do settle this question. – Jyrki Lahtonen Mar 27 '14 at 16:54
2

To bridge the tiny gap: For $n>2$, we have $2^n>2n$, and the second link by Martin Sleziak proves that all irreducible factors have degree dividing $2n$. – ccorn Mar 27 '14 at 18:35
Thank you very much! And I am sorry for this duplicate question. – Khako Mar 28 '14 at 05:59

0 Answers0