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The question is related to the motivation of Galois theory, which says given a quintic polynomial or beyond, whether one can solve it by radical depends on the solvability of the group of symmetries of the zeros of that polynomial (known as Galois group).

My question:

  • if we don't know the zeros of a polynomial explicitly, then how can we study symmetries in between them?

  • on the other hand, if we knew the zeros then why would study Galois theory, as we already know the zeros exists?

This is something I am confused. It seems to me that one can study symmetries of the zeros of a polynomial without knowing the zeros explicitly because given a polynomial over a field, the polynomial split in some extension of the base field.

I would appreciate to clarify the matter.

Learner
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    Not sure what you are asking. Of course the roots exist as complex numbers. And we might hope to understand the Galois group of a particular quintic (or of any polynomial) without needing to write the roots out explicitly. For instance, for a cubic, we can decide if the Galois group is $S_3$ or cyclic of order $3$ just be looking at the discriminant. No need to consider the explicit roots. – lulu Mar 12 '25 at 23:49
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    Check this out, for example. – anankElpis Mar 13 '25 at 00:02
  • @lulu, That is exactly what I wanted to know. So the upshot is that we can talk about solvability of the Galois group without knowing the zeros of a polynomial explicitly. And that is why Galois theory important – Learner Mar 13 '25 at 00:11
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    Also note that Galois theory has many more use cases besides finding roots of polynomials. (for example, the impossibility of classic geometry problems like squaring the circle, trisecting an angle or doubling a cube with straightedge and compass) – anankElpis Mar 13 '25 at 00:17
  • One symmetry is that all roots of an irreducible polynomial are algebraically indistinguishable, but once we select one of them, the others may or may not lie in the field containing the selected root. Considering roots as abstract objects lying in some splitting field helps a lot to learn about their symmetries. I made my foray into these topics by asking this question: https://math.stackexchange.com/q/2241608/72031 – Paramanand Singh Mar 13 '25 at 09:11

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