Mathematical connection between field theory and group theory
Lattice diagram of Q adjoin the positive square roots of 2 and 3, its subfields, and Galois groups.
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals.
Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle), and characterizing the regular polygons that are constructible (this characterization was previously given by Gauss but without the proof that the list of constructible polynomials was complete; all known proofs that this characterization is complete require Galois theory).
Galois' work was published by Joseph Liouville fourteen years after his death. The theory took longer to become popular among mathematicians and to be well understood.
Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.
In mathematics, Galoistheory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection,...
Galoistheory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in...
via Galois groups is called Galoistheory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups...
significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galoistheory. A result of Emil Artin...
mathematics, differential Galoistheory studies the Galois groups of differential equations. Whereas algebraic Galoistheory studies extensions of algebraic...
especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find...
Évariste Galois coined the term "group" and established a connection, now known as Galoistheory, between the nascent theory of groups and field theory. In...
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently...
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any...
In mathematics, topological Galoistheory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by Vladimir...
mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
F_{\infty }.} More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group. Let p...
group the Galois group of a Galois extension of the rational numbers? (more unsolved problems in mathematics) In Galoistheory, the inverse Galois problem...
problem of Galoistheory Given a group G, find an extension of the rational number or other field with G as Galois group. Differential Galoistheory The subject...
In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers...
among themselves. The significance of the Galois group derives from the fundamental theorem of Galoistheory, which proves that the fields lying between...
theory Fourier theoryGaloistheory Game theory Graph theory Grothendieck's Galoistheory Group theory Hodge theory Homology theory Homotopy theory Ideal...
intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galoistheory. Field extensions can be generalized to ring...
by introducing what is now called Galoistheory. Before Galois, there was no clear distinction between the "theory of equations" and "algebra". Since...
theory — Galoistheory — Game theory — Gauge theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory —...
higher do not have general solutions using radicals. Galoistheory, named after Évariste Galois, showed that some equations of at least degree 5 do not...
provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G...
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