[abstract algebra] is this ring isomorphic to

[nonequilibrium]

Homework Statement

Consider \frac{\mathbb Z_2[X]}{X^2+1}, is this ring isomorphic to \mathbb Z_2 \oplus \mathbb Z_2, \mathbb Z_4 or \mathbb F_4 or to none of these?

Homework Equations

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The Attempt at a Solution

- \mathbb F_4 No, because \mathbb Z_2[X] is a principle ideal domain (Z_2 being a field) and X²+1 is reducible in \mathbb Z_2[X] and in a principle ideal domain an ideal formed by a reducible element is not maximal and thus the quotient is not a field

- \mathbb Z_4 No, as it can obviously not be cyclical

- \mathbb Z_2 \oplus \mathbb Z_2 No. Because say there is an isomorphism \phi: \frac{\mathbb Z_2[X]}{X^2+1} \to \mathbb Z_2 \oplus \mathbb Z_2, then say \phi(X) = (a,b), then \phi(1) = \phi(X^2) = \phi(X)* \phi(X) = (a^2,b^2) = (a,b) since a and b are either zero or one. As a result "1" and "X" would have the same image. Contradiction.

Is this correct? If so, does this also seem like a good way to do it, or did I overlook an easier way to do this?

 

[micromass]

It seems to be correct! And you also used a very nice method to show this

If you don't want to work with the isomorphism in (3), then you could perhaps say that \mathbb{Z}_2\oplus\mathbb{Z}_2 has no nilpotent elements, while \mathbb{Z}[X]/(X^2+1) does.

 

[nonequilibrium]

Aha true :) thank you