Recent content by philbabbage

Is that actually sufficient for showing that the kernels are the same?

Yeaah, it's definitely awkward and needs some work. I'm still thinking about it: it has to be right, but the improvement isn't obvious to me. The kernel of phi is everything in (\mathbb{Z}/2\mathbb{Z})[x,y] that is zero in R/I. This is the same as all multiples of x^2+x+1, y^2+y+1, y^2+xy+x^2...

We know that if the images of phi and psi are isomorphic, then the kernels of each are equivalent. To me, this should imply that the number of canonical elements in each quotient are the same. In which case, I could argue that those polynomials less than z^2 + z + z are renamings like below: z...

Yes, I do. Let me change that. Actually, I think I do. Double check me, since I am quite new to both the subject and the notation. By \mathbb{Z}_2 I meant the integers mod 2 (although I realize I had written Z_2, which would definitely be weird), which I suspect is the same thing as...

Homework Statement Prove that (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is isomorphic to F_4[z]/(z^2 + z + 1) by showing that the kernel of \phi : (\mathbb{Z}/2\mathbb{Z})[x,y] \to (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is the...