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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 never prove that the alternating groups $A_n$ 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!

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No, he did not. You are used to see Galois theory from the modern point of view, developed by Emmy Noether and two of her students: van der Waerden and Emil Artin. I suggest that you read B. Melvin Kiernan's The Development of Galois Theory from Lagrange to Artin (Archive for History of Exact Sciences Vol. 8, No. 1/2, pp. 40–154).

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