Wantzel's proof used the abstract algebra of field extensions, a topic now typically grouped with Galois theory, but does not use the connection between field extensions and groups that is the subject of Galois theory itself.
Wantzel's proof, restated in modern terminology, uses the abstract algebra of field extensions, a topic now typically grouped with Galois theory. However Wantzel published these results earlier than Galois (published in 1846) and did not use the connection between field extensions and groups that is the subject of Galois theory itself.