# Finding an irreducible polynomial in Z [x] but reducible in F2 [x]

Hi everyone, I've been trying to find an irreducible polynomial in Z [x] but reducible in F2 [x].

I tried using this code:

i = 1
a = 0
while a == 0:
R = GF(2) ['x']
for p in R(x)polynomial(i)
i = i + 1
if not p.is_irreducible():
R.change_ring(ZZ)
if p.is_irreducible():
print p
a = 1


As I couldn't think of a way of finding an irreducible in Z and after that changing it to F2 I decided to do it backwards.

I'm in your graceful hands and sorry for my broken English.

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Homework ?

( 2021-11-17 18:23:19 +0100 )edit
1

I am not going to lie to you. Yeah it is.

( 2021-11-17 23:02:15 +0100 )edit
1

Hint : you are looking for a polynomial that you can factor in $\mathbb{F}_2$ but not in $\mathbb{Z}$. This might give you ideas :

sage: Pz.<z>=ZZ[]
sage: P2.<t>=GF(2)[]
sage: (z+1)^2
z^2 + 2*z + 1
sage: (t+1)^2
t^2 + 1

( 2021-11-18 17:43:32 +0100 )edit

Thank you, it helped a lot.

( 2021-11-28 14:10:36 +0100 )edit