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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Isomorphic Subfields

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