since I'm a bad algebraist, I have to relearn field theory. In this process, I came across the following concept:
For two field extensions $L \supset K$, $M \supset K$ there may be a $K$-isomorphism $L \to M$ (i.e. $L$ and $M$ are isomorphic via an isomorphism that leaves elements of $K$ fixed). I'm learning from van der Waerden's book (since it's one of the best on field theory) and he calls $L$ and $M$ equivalent field extensions. But I couldn't find this term with this meaning anywhere in the internet (neither in English nor in my mother tongue German).
So, since it's quite an old book, I thought maybe that there is now a new term for this, and I would be glad if anybody could help me out.
There is a category of field extensions of a field $k$ whose objects are field extensions of $k$. In your case, you have two objects $L$ and $M$ of this category, and an isomorphism in the category.