I have seen this terminology in two texts and for whatever reason cannot find a source for what this means. Does this mean the two extensions are field isomorphic? Thanks.
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1You cannot find a source? Search stackexchange next time, e.g., here. – Dietrich Burde Sep 22 '21 at 16:41
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Apologies. Understood if this question should be closed, but I spent a decent bit looking and didn't find anything. – Moni145 Sep 22 '21 at 16:43
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1No problem. I think one also has to get experience in searching. I found it rather quickly. You can try it next time. – Dietrich Burde Sep 22 '21 at 16:44
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If $L_1$ and $L_2$ are two extensions of $K$ then they are called $K$-isomorphic if there is an isomorphism of fields $\varphi: L_1\to L_2$ such that $\varphi(k)=k$ for all $k\in K$. Equivalently, this means that $L_1, L_2$ are isomorphic as $K$-algebras.
Note that in the special case $K=\mathbb{Q}$ this just means there is an isomorphism of fields between $L_1$ and $L_2$. (since any homomorphism leaves the elements of $\mathbb{Q}$ fixed)
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