Galois Group

Definition

A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial and the corresponding field extensions. It consists of automorphisms of a field extension that fix the base field, providing deep insights into the relationship between field theory and group theory.

5 Must Know Facts For Your Next Test

1. The order of a Galois group equals the degree of the corresponding field extension, linking algebraic properties directly to symmetry.
2. If a polynomial is separable, its Galois group acts transitively on its roots, showing that the roots can be permuted in a structured way.
3. The Fundamental Theorem of Galois Theory establishes a correspondence between subgroups of the Galois group and intermediate fields in the extension.
4. Galois groups provide criteria for determining whether a polynomial can be solved by radicals, revealing insights into solvable groups.
5. The Galois group can be computed explicitly for certain types of polynomials, such as quadratic or cubic equations, helping to visualize their symmetry.

Review Questions

• How does the Galois group relate to field extensions and the roots of polynomials?
  • The Galois group is intimately connected to field extensions as it describes the symmetries of the roots of a polynomial within those extensions. Specifically, for a given polynomial, its Galois group consists of all automorphisms that keep the base field fixed while permuting the roots. This relationship shows how the algebraic properties of polynomials are reflected in group structures, allowing us to study root behavior through symmetry.
• Discuss how the Fundamental Theorem of Galois Theory connects subfields with subgroups of a Galois group.
  • The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between subfields of a given field extension and subgroups of its Galois group. Specifically, each intermediate field corresponds to a subgroup where larger groups correspond to smaller subfields. This theorem provides powerful insights into how algebraic relationships manifest in both field and group structures, illustrating how understanding one can illuminate properties of the other.
• Evaluate the implications of Galois groups on the solvability of polynomials by radicals, particularly in relation to unsolvable cases like the general quintic.
  • Galois groups play a crucial role in determining whether a polynomial can be solved by radicals through their structural properties. For instance, if the Galois group is solvable, then the polynomial can typically be expressed using radicals. In contrast, for polynomials like the general quintic whose Galois group is not solvable, we find that there are no solutions expressible in terms of radicals. This highlights how deep connections between algebraic structures can dictate fundamental limitations in solving equations.

Related terms

Field Extension:

A field extension is a bigger field that contains a smaller field, allowing for the study of algebraic structures and their properties.

Automorphism:

An automorphism is a bijective map from a mathematical object to itself that preserves its structure, such as addition and multiplication in a field.

Solvable Group: A solvable group is a group that can be broken down into a series of abelian groups, indicating that its structure is simpler and often relates to the solvability of polynomials.