# Isomorphic Subfields

1. Jan 21, 2014

### Mandelbroth

I've got this weird thought in my head that isomorphic subfields of any field are equal. I'm having trouble believing this and I'm trying to come up with a counterexample to quell my head. But, I'm having a lot of trouble coming up with such a counterexample, and I don't know why.

Can anyone think of a quick counterexample?

So far, I've thought of the field isomorphic to $\mathbb{Z}/(2)$ generated by $0,1\in \mathbb{F}_4$. But, when I thought about it more, no other subfield is isomorphic to $\mathbb{Z}/(2)$ in $\mathbb{F}_4$, since $0$ and $1$ would necessarily be in the subfield. Then I thought about how $\mathbb{R}\cong \mathbb{R}i$ as vector spaces, but clearly not as rings ($\mathbb{R}i\not\ni 1$).

As usual, help is greatly appreciated.

2. Jan 21, 2014

### jgens

Notice that the inclusion k(x2,x3,...) ⊂ k(x1,x2,x3,...) is proper, yet these fields are isomorphic.