Definition

Noun: - A branch of abstract algebra that applies group theory to study the solutions (roots) of polynomial equations. It establishes a connection between field theory and group theory, providing a deep understanding of which algebraic equations can be solved by radicals (using a finite sequence of operations like addition, subtraction, multiplication, division, and taking roots).

Usage

• Galois theory is used to determine the solvability of polynomial equations.
• The fundamental theorem of Galois theory establishes a one-to-one correspondence between intermediate fields of a field extension and subgroups of its Galois group.

Examples

Advanced Usage

• Inverse Galois problem: A question within Galois theory asking whether every finite group can be realized as the Galois group of some Galois extension of the rational numbers.
• Differential Galois theory: An extension of classical Galois theory to differential equations, studying the solvability of differential equations in terms of elementary functions and integrals.

Variants and Related Words

• Galois group (n): For a given polynomial, the group of field automorphisms of its splitting field that leave the base field fixed. This group encodes the symmetries among the roots of the polynomial.
• Galois extension (n): A field extension that is both normal and separable, which is the central object of study in Galois theory.
• Galois correspondence (n): The fundamental theorem linking subfields and subgroups in a Galois extension.

Synonyms

• (There are no direct single-word synonyms. It is a specific, named theory. Descriptive phrases include:)
  • Application of group theory to equations
  • Algebraic solvability theory

Related Phrases

• Fundamental theorem of Galois theory: The core result stating the precise correspondence between the lattice of intermediate fields of a Galois extension and the lattice of subgroups of its Galois group.
• Solvable by radicals: A property of a polynomial equation that can be determined by examining whether its Galois group is a solvable group.

Noun

1. group theory applied to the solution of algebraic equations