Are the fields $\mathbb{Q}(i)$ and $\mathbb{Q}(2i)$ isomorphic? I'm confused since they seem to be equal as sets but $\mathbb{Q}(i)\cong \mathbb{Q}[X]/(X^2+1)$ but $\mathbb{Q}(2i)\cong \mathbb{Q}[X]/(X^2+4)$.
Are these fields isomorphic?
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field-theory
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0It's easier to see if you don't use the same symbol, $X$. You want to show that $\mathbb Q(X)/(X^2+1) \cong \mathbb Q(Y)/(Y^2+4)$ – 2012-09-04
2 Answers
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They are the same field.
Clearly $\mathbb{Q}(2i) \subset \mathbb{Q}(i)$
Note that $i \in \mathbb{Q}(2i)$ since $i = \frac{1}{2}(2i)$. Since $i \in \mathbb{Q}(2i)$, one has that $\mathbb{Q}(i) \subset \mathbb{Q}(2i)$.
Hence $\mathbb{Q}(2i) = \mathbb{Q}(i)$.
If they are equal as sets then they are certainly isomorphic. It is some sense better than being isomorphic since they are actually the same thing.
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The identity map is certainly an isomorphism :)