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Sep 22, 2020 at 16:09 comment added Marc Thanks! This argument is quite famous and I have actually seen this argument before, so I should have known that what I was trying to do had no chance of ever working...
Sep 22, 2020 at 15:07 comment added Will Sawin $J$ certainly preserves the lattice because it extends to an endomorphism over the complex numbers. So indeed $F$ must not.
Sep 22, 2020 at 14:52 history answered David E Speyer CC BY-SA 4.0