GALOIS THEORY Meaning and
Definition
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Galois Theory is a branch of mathematics that concerns itself with the study of field extensions and their associated groups. It was named after the French mathematician Évariste Galois, who made significant contributions to the field in the 19th century.
In Galois Theory, a field extension is an extension of one field by another, i.e., when one field is contained within another. The theory specifically focuses on the study of finite field extensions, where the field extension has a finite degree. Galois Theory seeks to understand the relationship between the field extension and the corresponding group of automorphisms, known as the Galois group.
The central idea behind Galois Theory lies in the fundamental theorem of Galois Theory, which establishes a correspondence between field extensions and subgroups of the Galois group. This correspondence provides deep insights into the solvability of polynomial equations by radicals. It allows us to determine whether a polynomial equation can be solved algebraically using only the operations of addition, subtraction, multiplication, division, and taking roots.
Furthermore, Galois Theory has important applications in other areas of mathematics, such as algebraic geometry and number theory. It provides a powerful framework for studying the symmetries and structure of mathematical objects, leading to a better understanding of their properties.
In summary, Galois Theory is a branch of mathematics that investigates the interplay between field extensions and their associated groups, aiming to unravel the solvability of polynomial equations and providing insights into various mathematical disciplines.
Common Misspellings for GALOIS THEORY
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Etymology of GALOIS THEORY
The word "Galois Theory" is named after the French mathematician Évariste Galois. The term "Galois Theory" itself was coined after his death in 1832, as a tribute to his groundbreaking work in the field. Galois made significant contributions to algebraic theory, particularly in the study of polynomial equations and their solvability by radicals. His work laid the foundation for what is now known as Galois Theory, which deals with the relationship between field extensions and solvability of polynomial equations.
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