- #1

kobulingam

- 10

- 0

**Z_2**

__/<u^4+u+1> isomorphism Z_2____/<u^4+u^3+u^2+u+1>__

How to figure an isomorphism from

Z_2## Homework Statement

How to figure an isomorphism from

Z_2

__/<u^4 + u +1> to Z_2____/<u^4 + u^3 + u^2 + u + 1>__

What I can now show (after a page and a half of work) is that the two polynomials generating the ideals are irreducible over Z_2.

I've been able to prove that the elements creating the ideas are both irreducible polynomials.

I can show proof that the ideals are irreducible, but I don't think we need to reuse that part in remaining solution. Essentially u^4 + u +1 has no linear factors by factor theorem (neither 0 nor 1 root), so only possibility is that it could be factored into 2 irreducible quadratics, and there is only once such quadratic in Z_2What I can now show (after a page and a half of work) is that the two polynomials generating the ideals are irreducible over Z_2.

## Homework Equations

I've been able to prove that the elements creating the ideas are both irreducible polynomials.

## The Attempt at a Solution

I can show proof that the ideals are irreducible, but I don't think we need to reuse that part in remaining solution. Essentially u^4 + u +1 has no linear factors by factor theorem (neither 0 nor 1 root), so only possibility is that it could be factored into 2 irreducible quadratics, and there is only once such quadratic in Z_2

__. Squared this quadratic and didn't get u^4 + u +1. Thus u^4 + u +1 is irreducible.__

Similarly, u^4 + u^3 + u^2 + u + 1 has no linear factors (neither 0 nor 1 is a root), so only possibility is that it's the product of an irreducible quadratic and irreducible cubic. There are only 2 possible such cubics. Multiplying each of these cubics with the irreducible quadratic does not give u^4 + u^3 + u^2 + u + 1. Thus u^4 + u^3 + u^2 + u + 1 irreducible over Z_2Similarly, u^4 + u^3 + u^2 + u + 1 has no linear factors (neither 0 nor 1 is a root), so only possibility is that it's the product of an irreducible quadratic and irreducible cubic. There are only 2 possible such cubics. Multiplying each of these cubics with the irreducible quadratic does not give u^4 + u^3 + u^2 + u + 1. Thus u^4 + u^3 + u^2 + u + 1 irreducible over Z_2

__. I am guessing this is the easy part of the answer, yet this itself stretched me fully...__