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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 $\mathbb{Q}(2i)\cong \mathbb{Q}[X]/(X^2+4)$.

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    They are isomorphic; the isomorphism just doesn't send the first $X$ to the second $X$. (It would be a good idea to use a different letter in the second presentation for this reason.)2012-09-04
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    It'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

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