I've been reading Galois' First Memoir, where he introduces Galois Theory by giving a sufficient and necessary condition for a polynomial to be solvable by radicals. The proofs are a bit sketchy and the terminology is strange, which I'm guessing is normal. Up to this point I have seen no mention of the concept of a basis, which I deem rather interesting since bases are an essential, even necessary pillar of Galois Theory.
However I've read that Galois was able to prove some of his theorems by use of different strategies as the ones taught today, for instance, Peter M. Neumann explains that Galois did ever prove the alternating groups An are simple for n<4
, supposedly, he did not need to.
Which leads me to wonder: Did Galois make use of bases to prove his Fundamental Theorem? If so, could someone please point me to the source. And if not, how was he able to prove there was a bijection between the intermediate fields of an extension and the subgroups of its Galois group?
I appreciate any thoughts/help!