Degree of an extension

Definition

The degree of an extension is the dimension of a field extension as a vector space over its base field. This concept is crucial in understanding the structure and properties of field extensions, which can be algebraic or transcendental, and plays a significant role in distinguishing between simple and multiple extensions, as well as understanding their implications in Galois theory and algebraic number theory.

5 Must Know Facts For Your Next Test

1. The degree of an extension can be finite or infinite; finite degrees indicate that the extension has a finite basis over the base field.
2. If K is a field and L is an extension of K, the degree [L : K] is equal to the number of elements in any basis of L over K.
3. For simple extensions generated by a single element, such as L = K(α), the degree can be computed using the minimal polynomial of α over K.
4. The degree is multiplicative for tower extensions: if you have extensions K ⊆ L ⊆ M, then [M : K] = [M : L] * [L : K].
5. In Galois theory, if an extension is normal and separable, its degree corresponds to the size of its Galois group.

Review Questions

• How does the degree of an extension relate to vector spaces, and why is this important for understanding field extensions?
  • The degree of an extension relates to vector spaces by defining it as the dimension of the extended field considered as a vector space over its base field. This relationship is important because it helps identify how many elements are needed to form a basis for the extended field. Understanding this dimension sheds light on the complexity and structure of the field extension, informing properties such as algebraic independence and polynomial roots.
• Discuss how the degree of an extension influences whether elements are algebraic or transcendental.
  • The degree of an extension directly influences whether elements are classified as algebraic or transcendental. An element in an extension has finite degree if it is algebraic over the base field, meaning it satisfies a polynomial equation with coefficients from that field. Conversely, transcendental elements do not satisfy any such polynomial equation, leading to an infinite degree. Understanding these distinctions clarifies how different elements behave within extensions and aids in exploring their implications in algebraic structures.
• Evaluate the significance of degrees in Galois theory, particularly regarding normal and separable extensions.
  • In Galois theory, degrees play a significant role in analyzing normal and separable extensions. For such extensions, the degree corresponds to the size of the Galois group associated with them, which encapsulates all automorphisms fixing the base field. This connection between degrees and Galois groups not only reveals symmetries within the extensions but also provides insights into solvability by radicals and other properties crucial for understanding more complex algebraic structures.

Related terms

Field extension: A field extension is a pair of fields where one field contains the other as a subfield, allowing for operations and properties to be explored in a broader context.

Algebraic element: An algebraic element is an element of a field extension that is a root of a non-zero polynomial with coefficients from the base field.

Galois group: The Galois group of a field extension is the group of field automorphisms that fix the base field and helps in analyzing the symmetries and properties of the extension.