Normal extension

Definition

A normal extension is a type of field extension where every irreducible polynomial in the base field that has at least one root in the extension splits completely in that extension. This means that all roots of these polynomials are included within the extension. Normal extensions are key to understanding Galois theory, as they help identify when a field extension can be associated with a Galois group and thus have a well-defined structure.

5 Must Know Facts For Your Next Test

1. Normal extensions ensure that if a polynomial has a root in the extension, all of its roots are also included, which is crucial for working with Galois groups.
2. Every Galois extension is a normal extension, but not every normal extension is necessarily Galois; it must also be separable.
3. An example of a normal extension is the splitting field of an irreducible polynomial over a field, where all roots are contained in the extension.
4. Normal extensions relate closely to the concept of separability; if an irreducible polynomial has distinct roots, its splitting field is a normal extension.
5. The Fundamental Theorem of Galois Theory links the structure of normal extensions with the properties of their corresponding Galois groups.

Review Questions

• How does a normal extension relate to the concept of irreducible polynomials and their roots?
  • A normal extension is defined by its relationship with irreducible polynomials. Specifically, if an irreducible polynomial in the base field has at least one root in the normal extension, then it must have all its roots in that extension as well. This property ensures that all symmetries associated with the roots can be fully understood within the framework of Galois theory.
• What distinguishes a normal extension from a general algebraic extension, and why is this distinction important?
  • While both normal extensions and general algebraic extensions arise from adjoining roots of polynomials, normal extensions specifically require that every irreducible polynomial with a root in the extension must split completely within it. This distinction is crucial because it allows for a clearer understanding of Galois groups and their structure, as normal extensions are necessary for defining Galois connections accurately.
• Evaluate how normal extensions contribute to solving polynomial equations through Galois theory.
  • Normal extensions play a pivotal role in solving polynomial equations by facilitating the application of Galois theory. In particular, they provide the necessary framework for linking the structure of field extensions with the solutions to polynomials. By ensuring that every irreducible polynomial splits completely, these extensions allow us to analyze symmetries among roots using Galois groups, thus paving the way for determining solvability criteria for various classes of equations.

Related terms

Galois group: The group of field automorphisms of a normal extension that fix the base field, representing symmetries in the roots of polynomials.

Splitting field: The smallest field extension in which a polynomial splits into linear factors, thereby providing all the roots of the polynomial.

Algebraic extension: A field extension created by adjoining roots of polynomials from the base field, which can include both normal and non-normal extensions.