To be fair I think the idea of using algebraic number theory to approach the problem had been tried before (Tsimerman mentions he tried a similar approach that the model ultimately succeeded with, but didn’t persist with it.) It’s quite a general trick to use algebraic number theory for constructions in the plane, as you have the lattice associated with the ring of integers of number fields.
I personally am blown away by the proof but it would be far more impressive had it come up with a novel connection between fields, or indeed if it had turned out there wasn’t a counterexample and it proved a tight upper bound (See Gowers’ initial reaction.)
To be fair I think the idea of using algebraic number theory to approach the problem had been tried before (Tsimerman mentions he tried a similar approach that the model ultimately succeeded with, but didn’t persist with it.) It’s quite a general trick to use algebraic number theory for constructions in the plane, as you have the lattice associated with the ring of integers of number fields.
I personally am blown away by the proof but it would be far more impressive had it come up with a novel connection between fields, or indeed if it had turned out there wasn’t a counterexample and it proved a tight upper bound (See Gowers’ initial reaction.)