I don't think this is Galois theory. Galois theory is about "The Fundamental Theorem of Galois Theory", which states that there is a nice mapping from field-extensions to group-extensions, where the resulting groups are usually finite. When the resulting groups are finite (as they usually are), many problems involving field extensions can be solved using brute-force search.

Galois fields happen to be something else named in honour of Galois.