Finite Extension

Definition

A finite extension is a field extension in which the larger field has a finite dimension as a vector space over the smaller field. This concept connects various aspects of algebraic structures, showcasing how algebraic numbers and integers can form fields with finite degrees, and how properties such as norms, traces, and discriminants are integral to understanding these extensions.

5 Must Know Facts For Your Next Test

1. A finite extension can be either algebraic or transcendental, but in most contexts related to number theory, we focus on algebraic finite extensions.
2. The degree of a finite extension provides critical information about the extension's structure and can be calculated using the minimal polynomial of any element in the extension.
3. Finite extensions often exhibit specific behaviors with respect to norms and traces, which are used to study the properties of elements within the extension.
4. In number theory, every number field is a finite extension of the rational numbers, demonstrating that they have a well-defined structure in terms of dimensionality.
5. The discriminant of a finite extension helps determine the ramification properties of primes in relation to that extension.

Review Questions

• How does the degree of a finite extension relate to the structure of algebraic numbers and integers?
  • The degree of a finite extension directly measures how 'large' the extension is in terms of vector space dimensions. For example, when considering algebraic numbers, each number field represents a finite extension of the rationals, where the degree signifies how many basis elements are needed to express elements of the field. Thus, understanding this dimension aids in grasping the complexity and relationships among different algebraic numbers and integers.
• Discuss the significance of norm and trace maps in finite extensions and their implications for field theory.
  • Norm and trace maps play crucial roles in understanding the behavior of elements within finite extensions. The norm is a multiplicative function that provides insights into how elements scale when considered over the base field, while the trace gives information about sums of roots related to those elements. These maps not only help determine algebraic properties but also link to practical calculations involving discriminants and Galois groups.
• Evaluate how finite extensions influence the behavior of primes within number fields and their decompositions.
  • Finite extensions significantly impact prime factorization within number fields through their decompositions into prime ideals. When moving from a base field to a finite extension, primes may split into several factors, remain inert, or ramify. This behavior is essential for understanding how different primes relate across fields and affects computations involving local properties like valuations or completions, showcasing the intricate connections between number theory and algebraic structures.

Related terms

Algebraic Closure: The smallest field extension of a given field in which every non-constant polynomial has a root.

Degree of an Extension: The dimension of a field extension as a vector space over the base field, which indicates how 'large' the extension is.

Integral Closure: The set of all elements in a field that are integral over a ring, providing insight into the relationship between algebraic integers and number fields.