The statement in the title is true for any field $K$ (as far as I know), and the proof wasn't that hard (as far as I know again). After I read about the sketch of the proof of Ax-Grothendieck theorem from this answer, I wonder if it is possible to prove the statement in a similar way, because proving it for finite fields is trivial by considering the size of groups. I think it may not be possible since such argument only seems to work for algebraically closed fields, but I'm not an expert of model theory so I think that I might be wrong.
2026-04-07 05:28:41.1775539721
Model-theoretic proof for $\mathrm{GL}_{n}(K) \simeq \mathrm{GL}_{m}(K) \Rightarrow n = m$
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