What are some practical and potential applications of Galois theory in different field of mathematics?

Question

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？

Answer 1

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.