Z_2 /<u^4+u+1> isomorphism Z_2 /<u^4+u^3+u^2+u+1>

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In summary, an isomorphism from Z_2[u]/<u^4 + u +1> to Z_2[v]/<v^4 + v^3 + v^2 + v + 1> can be figured by showing that the two polynomials generating the ideals are irreducible over Z_2 and using the theorem that tells us something about all possible homomorphisms in this case.
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kobulingam
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Z_2/<u^4+u+1> isomorphism Z_2/<u^4+u^3+u^2+u+1>

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.



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_2. I am guessing this is the easy part of the answer, yet this itself stretched me fully...
 
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  • #2


kobulingam said:
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.
Er, you mean two irreducible quadratics, don't you?


Anyways, don't you know any theorems (or can comptue one) that tell you something about all possible homomorphisms [itex]R[x] / \langle f(x) \rangle \to S[/itex], where R and S are rings?

(If you need a hint, first consider [itex]R[x] \to S[/itex])


Incidentally, a useful syntactic tip is to use different indeterminate variables in your two different rings. I.E. write them as
Z_2/<u^4 + u +1>

and
Z_2[v]/<v^4 + v^3 + v^2 + v + 1>.​
 

1. What is the meaning of "Z_2 / isomorphism Z_2 /"?

"Z_2" refers to the field of integers modulo 2, and "/" and "/" represent two different polynomial rings. The statement "Z_2 / isomorphism Z_2 /" means that there exists a bijective mapping between the elements of these two polynomial rings, preserving their algebraic structure and operations.

2. What is the significance of this isomorphism in algebraic structures?

This isomorphism is significant because it allows us to study the properties and behaviors of one algebraic structure (Z_2 /) by looking at the corresponding structure (Z_2 /) which may be easier or more familiar to work with. It also provides a way to generalize results from one structure to another.

3. How can we prove the existence of this isomorphism?

To prove the existence of this isomorphism, we need to show that there exists a bijective mapping between the elements of Z_2 / and Z_2 / that preserves their algebraic operations. This can be done by explicitly defining the mapping and showing that it satisfies the necessary properties.

4. Can this isomorphism be extended to other polynomial rings and fields?

Yes, this isomorphism can be extended to other polynomial rings and fields as long as they have similar algebraic structures and operations. This is because the isomorphism is based on the properties of the polynomial rings and not on specific elements or coefficients.

5. What are the practical applications of this isomorphism?

This isomorphism has practical applications in various fields such as coding theory, cryptography, and error correction. It also has applications in studying and solving problems related to polynomial rings and fields in areas such as abstract algebra and number theory.

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