The I. N. Herstein text does Galois theory with fully precise proofs. Learn those and could see how to "invent" Galois theory yourself. As I recall Herstein do

The I. N. Herstein text does Galois theory with fully precise proofs. Learn those and could see how to "invent" Galois theory yourself. As I recall Herstein does not cover Gröbner basis. By now don't really have to look for a book and, instead, can use the Internet -- yes, some Internet material does provide careful proofs.