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.
2026-05-04 14:53:07.1777906387
What does it mean for two field extensions over $\mathbb{Q}$ to be "$\mathbb{Q}$-isomorphic?"
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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)