Make a list of all irreducible polynomials of degrees 1 to 5 over F2

[Hill]

TL;DR

F2 = (0, 1)

To this exercise,

the textbook's solution is

I didn't follow their method and have found another degree 5 solution, $X^5+X^4+X^2+X+1$.
Is it wrong or the book has omitted this solution?

 

[martinbn]

TL;DR Summary: F2 = (0, 1)

To this exercise,
View attachment 355323
the textbook's solution is
View attachment 355324
I didn't follow their method and have found another degree 5 solution, $X^5+X^4+X^2+X+1$.
Is it wrong or the book has omitted this solution?

It is irreducible. I guess they missed it.

 

[Hill]

It is irreducible. I guess they missed it.

As well as
$X^5+X^4+X^3+X+1$
$X^5+X^3+X^2+X+1$
$X^5+X^4+X^3+X^2+1$
(?)

 

[Gavran]

As well as
$X^5+X^4+X^3+X+1$
$X^5+X^3+X^2+X+1$
$X^5+X^4+X^3+X^2+1$
(?)

Yes.
There are six of them.

 

[mathwonk]

This seems a little easier than the book's hints make it. A poly. over Z/2 is divisible by X or X+1 iff it either ends in X or has an even number of terms, hence the irreducible cubics end in 1 and have an odd number of terms, i.e. X^3+X^2 + 1, and X^3+X+1. A reducible quartic or quintic either ends in X or has an even number of terms or is divisible by X^2+X+1....It is still somewhat tedious, but maybe not as long as multiplying together all those factors to compute all reducible examples. It is interesting that after all these years I still have trouble checking irreducibility. I guess it is just inherently difficult. Such problems have been challenging for years in algebraic geometry: Severi's conjecture, irreducibility of moduli of curves, components of Hilbert schemes,....