Apart from providing solution to classical questions like
Solvability of quintic or higher degree polynomial equations in terms of radicals
Also, any related book recommendation, guys？
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7Number theory, especially algebraic number theory and arithmetic geometry use Galois theory. – Lukas Heger Oct 13 '23 at 11:11
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2Galois theory can be restated using category theory, so you might be interested in taking a look at this. – rschwieb Oct 13 '23 at 11:30
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Galois theory is a general phenomenon, that shows up in the context of Category Theory, Mathematical Logic, Differential Equations, Algebraic Topology and Algebraic Geometry. Here is a summary of two most relevant examples of Galois theory itself, from number theory and algebraic geometry separately.
- Class Field Theory: Given a number field $K$, its class group $Cl(K)$ measures how much its integer ring $\mathcal O_K$ fails to have unique factorization property. This purely intrinsic group is isomorphic to a Galois group $G(L|K)$ where $L$ is a certain abelian extension of $K$, that is called the Hilbert class field of $K$. This is just the starting point of Galois theory being an essential part of modern algebraic number theory.
- Weil's Conjectures: The rational points on an algebraic variety can be viewed as those that are invariant under Galois action of $G(\overline{K}|K)$. When the base field is a finite field $\mathbb F_q$, the Galois group is essentially generated by the Frobenius map. To count the fixed points of the map, one can invoke ideas of counting fixed points in topology.
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