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So I understand Galois theory mainly from the point view of root fields. I am not quite sure that did I get everything right. So if we have root field extension K of F, then the Galois group Gal(K:F) is those automorphisms of K that keep F fixed. These automorphisms correspond to different permutations of roots of the polynomial which root field K is. And also every subgroup of Gal(K:F) determines an intermediate field between K and F.

So with Galois theory, you can say something about a polynomial with permutations of it's roots. Finding the right needed properties of polynomials Galois group we can solve which polynomials are solvable in radicals and which aren't. This leads to the lack of a quintic formula or any higher.

Did I get right? Any answers are appreciated :)

Touko Puro
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  • We just had this question here. An important point was, that not every permutation need lead to an automorphism. Does this answer your question? – Dietrich Burde Apr 29 '20 at 12:01
  • Yes I knew that not all permutations of roots lead to automorphisms. Just wanted to check that did I understand everything correctly. – Touko Puro Apr 29 '20 at 12:11
  • Yes, so far it looks good. But Galois theory is much more than what you have listed. We have a bijective correspondence between intermediate fields and subgroups, i.e., we can understand (Galois) field extensions by group theory. Also, it appears in number theory and topology, see for example this post. – Dietrich Burde Apr 29 '20 at 12:18

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