>
> I had actually started down this road, managing to prove that the 
> algebraic numbers were a field via linear combinations of powers


Are you going to contribute that proof, though? As far as I know it hasn't 
been formalized in set.mm yet. Also your proof method seems especially 
interesting, for instance, all proofs I've heard of either use symmetric 
polynomials of Hamilton-Cayley theorem. 

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