Algebraic extension

Definition

An algebraic extension is a type of field extension where every element of the extended field is a root of a non-zero polynomial with coefficients from the base field. This means that when you have a field and you create a new field by adding elements that satisfy certain polynomial equations, you end up with an algebraic extension. These extensions play a key role in understanding the structure of fields, especially when discussing concepts like algebraic closures and how different field extensions relate to one another.

5 Must Know Facts For Your Next Test

1. Algebraic extensions can be finite or infinite, depending on whether they contain a finite or infinite number of elements not in the base field.
2. Every algebraic extension is contained within some algebraically closed field, meaning that you can always find a larger field where all polynomials can be factored completely.
3. A simple algebraic extension can be formed by adjoining a single algebraic element to a base field, creating a new field that is still manageable to work with.
4. The degree of an algebraic extension is defined as the dimension of the extension as a vector space over the base field, providing insights into the complexity of the extension.
5. Algebraic extensions are crucial for solving polynomial equations since they often contain roots that are not available in the base field.

Review Questions

• How does an algebraic extension enhance our understanding of polynomial roots within a given base field?
  • An algebraic extension allows us to include new elements that are roots of polynomials with coefficients from the base field. This means that by extending our field, we can solve polynomial equations that may not have solutions within the original field. As we introduce these new elements, we begin to see how various polynomial relationships evolve and how they impact the structure and properties of our extended field.
• Discuss how minimal polynomials relate to algebraic extensions and their significance in determining properties of these extensions.
  • Minimal polynomials play an essential role in defining algebraic extensions because they provide the simplest polynomial relationship for each element added to the base field. When you adjoin an element to form an algebraic extension, its minimal polynomial helps identify its degree and thus determines the overall structure of the extension. By studying these minimal polynomials, we gain insight into how elements interact and fit together within the larger framework of fields.
• Evaluate the implications of every algebraic extension being contained within some algebraically closed field and its effects on solving polynomials.
  • The fact that every algebraic extension is contained within some algebraically closed field means that we can always find a 'safe' space where all polynomials can be fully factored and solved. This containment not only helps us understand how complex our algebraic extensions are but also assures us that for any polynomial equation, there exists a solution in some larger context. This property leads to deeper insights into both number theory and algebra by showing how different fields interconnect through their roots and extensions.

Related terms

Field extension: A field extension is a bigger field that contains a smaller field as a subfield, allowing for the study of how these fields relate to each other.

Algebraic closure:

The algebraic closure of a field is an extension in which every non-constant polynomial has a root, making it the smallest algebraically closed field containing the original field.

Minimal polynomial: The minimal polynomial of an element over a field is the monic polynomial of least degree that has that element as a root, which helps in determining the structure of the algebraic extension.