Separable Extension

Definition

A separable extension is a field extension where every element can be expressed as a root of a separable polynomial, meaning that the minimal polynomial of each element does not have repeated roots. This concept is crucial for understanding the structure of field extensions and their relationships to Galois theory and algebraic equations.

5 Must Know Facts For Your Next Test

1. Separable extensions are essential in the context of Galois theory as they ensure that roots behave nicely under field automorphisms.
2. In characteristic zero, all field extensions are separable since polynomials have distinct roots in their splitting fields.
3. For finite fields, separable extensions must be considered carefully, as they can have inseparable components due to the presence of repeated roots.
4. Every finite separable extension has a degree equal to the number of its distinct embeddings into an algebraic closure.
5. The study of separable extensions plays a critical role in proving results such as Galois' criterion for solvability by radicals.

Review Questions

• How does the concept of separable extensions relate to the behavior of field automorphisms?
  • Separable extensions allow for a more straightforward application of field automorphisms because every element can be represented by distinct roots. This means that when you apply an automorphism, it can permute the roots without encountering repeated roots, which simplifies the structure of the Galois group. The lack of repeated roots ensures that these automorphisms act independently and maintain the integrity of the extension.
• Discuss how separable polynomials contribute to determining whether a field extension is separable and why this matters in Galois theory.
  • Separable polynomials are critical in identifying separable extensions because if all minimal polynomials over a field have distinct roots, then the extension formed by adjoining their roots is separable. This property ensures that the extension is well-behaved and allows for a clear understanding of its Galois group. In Galois theory, this distinction between separable and inseparable extensions helps classify fields and understand their symmetries regarding root structures.
• Evaluate the implications of having an inseparable extension in terms of solvability by radicals and how it contrasts with separable extensions.
  • Inseparable extensions complicate the analysis of solvability by radicals because repeated roots can lead to non-unique representations of elements, which disturbs the usual framework we rely on with separable extensions. For instance, an inseparable polynomial might lead to multiple embeddings that do not correspond neatly to distinct roots. In contrast, separable extensions ensure that we can uniquely determine how to express roots via radicals, thus facilitating Galois' criterion and revealing more about the solvability landscape within algebraic equations.

Related terms

Separable Polynomial:

A polynomial that has distinct roots in its splitting field, implying that its derivative is not identically zero.

Galois Extension:

A field extension that is both normal and separable, allowing for a well-defined Galois group that captures symmetries of the roots of polynomials.

Inseparable Extension: A field extension where at least one element's minimal polynomial has repeated roots, leading to complications in the behavior of roots under field automorphisms.