Degree of extension

Definition

The degree of extension refers to the dimension of a field extension as a vector space over its base field. It measures how many elements from the larger field can be expressed as linear combinations of elements from the smaller field. Understanding the degree of extension is crucial for exploring properties like algebraic closures, norms, traces, and various structures in number theory.

5 Must Know Facts For Your Next Test

1. The degree of an extension is defined as the dimension of the larger field when viewed as a vector space over the smaller field, often denoted as $[L:K]$ for fields $L$ and $K$.
2. If an extension is finite, then it has a degree which is a positive integer, while an infinite extension has an infinite degree.
3. For algebraic extensions, the degree can be determined by finding the minimal polynomial of elements in the extension over the base field.
4. In some cases, like with splitting fields, the degree can reveal information about how many roots a polynomial has and how they relate to each other.
5. Degree plays a key role in understanding concepts like ramification and how primes decompose in number fields.

Review Questions

• How does the degree of an extension influence the structure and properties of algebraic closures?
  • The degree of an extension indicates how many new elements are introduced when moving to an algebraic closure. This affects properties like solvability; if an extension has finite degree, it can sometimes lead to simpler structures in terms of finding roots for polynomials. As you extend fields to their algebraic closures, understanding the degree helps in grasping how these new roots interact with existing ones and how they might be organized.
• Compare the degree of finite extensions with infinite extensions and discuss their implications on algebraic structures.
  • Finite extensions have a well-defined degree which gives insights into their dimensionality as vector spaces over their base fields. This contrasts sharply with infinite extensions, which do not have a fixed dimension. The implications are significant; finite extensions can yield specific results in terms of norms and traces, while infinite extensions require different methods for analysis. Understanding this distinction helps in studying more complex algebraic structures.
• Evaluate how knowledge of the degree of extension is essential in analyzing decomposition and inertia groups within Galois theory.
  • Understanding the degree of an extension is crucial for analyzing decomposition and inertia groups because these groups characterize how primes split in number fields. The degree indicates how many distinct places correspond to a prime in the base field after extending it. This relationship is foundational when investigating ramification behavior as well; different degrees lead to different ramification scenarios, significantly influencing Galois theory's broader framework.

Related terms

Field extension: A field extension is a bigger field that contains a smaller field, allowing for additional elements and operations that aren't possible in the smaller field.

Algebraic closure: An algebraic closure of a field is a larger field in which every non-constant polynomial has a root, essentially ensuring that all algebraic equations can be solved.

Galois group: The Galois group of a field extension is a group that describes the symmetries of the roots of polynomials within that extension, directly related to the structure and properties of the extension.