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Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible polygons) ?

I understand that Galois Theory had been extended and generalized to many fields. Just wonder in those fields, had the theory solved any major problems ?

I asked this question is because, some times, "extending or generalize a theory" to a different field is not difficult, but does this "new theory" really solve problems ?

ben
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2 Answers2

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This is a very reasonable question. Indeed, not all "generalizations" are anything more than keeping-busy, or technical improvements. But, in fact, "Galois theory" is constantly invoked in algebraic number theory, and in algebraic geometry after Grothendieck, for example. It's not so much that it "solved big (identifiable, clearly demarcated) problems", but that it made various things make sense, have names, and be promised to behave consistently with intuition.

That is, it's not so much that there were sensational/heroic episodes due to Galois theory, but that mundane enterprises could operate at a higher level, so that now "Galois theory" is an implicit part of the context of very many algebra-oriented discussions. "It goes without saying..." Not dramatic, perhaps, but indispensable.

paul garrett
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Wiles' proof of Fermat's Last Theorem makes extensive use of Galois theory.

G. Schiele
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