Galois theory


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Ga·lois theory

 (găl-wä′)
n.
The part of algebra concerned with the relation between solutions of a polynomial equation and the fields containing those solutions. It gives conditions under which the solutions can be expressed in terms of addition, subtraction, multiplication, division, and of the extraction of roots.

[After Évariste Galois (1811-1832), French mathematician.]

Galois theory

(ˈɡælwɑː)
n
(Mathematics) maths the theory applying group theory to solving algebraic equations
[C19: named after Évariste Galois (1811–32), French mathematician]
ThesaurusAntonymsRelated WordsSynonymsLegend:
Noun1.Galois theory - group theory applied to the solution of algebraic equations
math, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement
group theory - the branch of mathematics dealing with groups
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This text introduces the theory of separable algebras over commutative rings, covering background on rings, modules, and commutative algebra, then the key roles of separable algebras, including Azumaya algebras, the henselization of local rings, and Galois theory, as well as AaAaAeAa[umlaut]ta algebras, connections between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups.
The attempt to extend the Galois theory for commutative algebras, due to Chase, Harrison and Rosenberg [29], to the case of partial group actions (see reference [36]) spawned a new and unexpected development, namely the extension of the notion of partial actions to the realm of Hopf algebras.
Moisil, and a few others, there were some isolated attempts of renewal, of incorporating, in university courses some achievements of mathematics in the 19th century and at the beginning of 20th century, such as the epsilon-delta analysis, crystallized by Cauchy, Riemann and Weierstrass, the Galois theory, the Cantor set theory, the theory of integral equations, modern logic, etc.
His textbook for graduate and undergraduate courses in number and Galois theory covers from Fermat to Gauss, class field theory, complex multiplication, and additional topics.
Other revisions to this second edition include an earlier introduction to noncommutative rings; a simpler treatment of the existence of free groups; and new discussions of Galois theory for infinite extensions, the normal basis theory, abelian categories, and module categories.
As a result of Lagrange's theorem and basic Galois theory, [[L.
It seems possible that the Argand image may point the way to a more surveyable (or visualisable) form of a very abstract theory: the Galois Theory (see, for example, Littlewood, 1965, p.
And in the chapters on Bichat (chapter 4), Davy (chapter 5), and Galois (chapter 6) that follow, Chai traces out analogous distinctions between degrees of reflexivity, ranging from Bichat's attempt to develop a new theory of vitality, to Galois' more ambitious field theory; Galois theory could be extended to include new members in a group that are not yet known--a group defined by a "principle of containment" rather than an account of its elements (147).
One of the nicest actual constructions of the 17-gon is Richmond's (1893), as reproduced in Stewart's Galois Theory.
This text introduces undergraduates with no knowledge of the subject to abstract algebra, covering group theory, ring theory, and fields and Galois theory.