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Someone could tell you that Galois theory is the study of field extensions and the structure of their automorphism groups but is that really going to help?



The fundamental theorem of Galois theory reduces certain problems in field theory (like finding roots of polynomials) to group theory (counting permutations and symmetries), which makes them simpler and easier to understand and solve (or prove non-solvable). Galois theory is the proof and applications of this theorem, and related topics. Key to this theory is being more methodical about extending the rational numbers into the real numbers, by introducing new numbers one at a time, instead of all at once.

One of the immediate discoveries in beginning this study is the fact that in many common cases you cannot add just one numbers one at a time, but must add 2 or more numbers at once. These sets of numbers are called conjugates, and have the interesting property that even though you can prove how many must exist and that they are distinct from each other, they are otherwise identical except in the arbitray names you give them.




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