Fixed field

Definition

A fixed field is a subfield of a given field extension that remains unchanged under the action of a particular group of field automorphisms. In the context of Galois theory, the fixed field corresponds to those elements in the larger field that are invariant under the transformations dictated by the Galois group. Understanding fixed fields is crucial as they provide insight into the relationship between the field extensions and the symmetries present in their structures.

5 Must Know Facts For Your Next Test

1. Fixed fields are essential in connecting Galois groups to field extensions, as every subgroup of a Galois group corresponds to a fixed field.
2. For any finite Galois extension, the fixed field can be characterized as being generated by elements that remain unchanged under the actions of all automorphisms in the Galois group.
3. The degree of the fixed field extension can be determined by the index of the subgroup corresponding to it within the Galois group.
4. If you have a splitting field of a polynomial, its fixed field will relate closely to the coefficients of that polynomial and how they transform under automorphisms.
5. In essence, studying fixed fields helps in analyzing how symmetries affect algebraic structures, making them integral to understanding both abstract algebra and number theory.

Review Questions

• How do fixed fields relate to Galois groups and their subgroups?
  • Fixed fields are directly tied to Galois groups through the correspondence established by the Fundamental Theorem of Galois Theory. Each subgroup of a Galois group corresponds to a specific fixed field; thus, analyzing these subgroups provides insight into the structure and properties of the associated field extensions. When you identify a subgroup, you can find its corresponding fixed field, allowing for deeper exploration into how elements behave under various automorphisms.
• Discuss how the concept of fixed fields aids in understanding polynomial roots in relation to their coefficients.
  • Fixed fields help illustrate how polynomial roots interact with their coefficients through automorphisms in a splitting field. When examining a polynomial and its splitting field, the fixed field includes elements that remain invariant under these transformations. This invariance is vital because it allows us to see which properties of the roots depend only on the coefficients, enhancing our grasp on root behavior and providing critical insights into solving polynomials within various field contexts.
• Evaluate how understanding fixed fields contributes to broader mathematical concepts such as solvability by radicals and symmetric functions.
  • Understanding fixed fields deepens our comprehension of broader mathematical ideas like solvability by radicals and symmetric functions. Fixed fields highlight which elements can be constructed from roots using algebraic operations and radicals, providing a pathway for determining whether certain equations can be solved in radical terms. Additionally, they connect to symmetric functions by showing how invariants under transformations affect polynomial expressions. Thus, this knowledge not only enriches our grasp of Galois theory but also its applications across algebra.

Related terms

Galois Group: The Galois group is the group of all automorphisms of a field extension that fix a base field, revealing how the roots of a polynomial relate to each other.

Field Extension: A field extension is a bigger field that contains a smaller field, allowing for the study of polynomials and their roots over different fields.

Normal Closure: The normal closure of a field extension is the smallest normal extension containing that extension, often useful in understanding fixed fields and their properties.