# Math Help - Maximal Ideal

1. ## Maximal Ideal

Let R = Z x Z = {(x,y)|x,y in Z}

Show that Z x 2Z is maximal in R. Where, Z x 2Z = {(x,y)| x in Z, y in 2Z}

2. quotient out by it you get $\mathbb{Z}_2$ which is a field, thus that ideal is maximal.

3. I know that Z/2Z = Z_2 (is a field) but how do you know that ZxZ/Zx2Z is a field?

4. Just take a look at the cosets. $(a,b)+(Z, 2Z)=(a+Z,b+2Z)=(Z,b+2Z)$. There are only two cases, if b is even, it is the 0, if b is odd, it is the other one. Thus it is a group with 2 elements, it is the finite field of order 2.