Fixed field

Definition

A fixed field is the subfield of elements in a field extension that remain unchanged under the action of a group of automorphisms. In the context of Galois theory, it plays a crucial role as it relates the structure of field extensions to their automorphisms, connecting important concepts such as Galois groups and normal extensions.

5 Must Know Facts For Your Next Test

1. The fixed field corresponds directly to the elements that are invariant under the action of automorphisms in a Galois extension.
2. Every finite group of automorphisms has a unique fixed field, which is an essential concept in understanding the Galois correspondence.
3. If you have a Galois extension, then the degree of the extension is equal to the order of its Galois group, linking these two structures through their fixed fields.
4. The Fundamental Theorem of Galois Theory states that there is a one-to-one correspondence between subgroups of the Galois group and intermediate fields, where each subgroup’s fixed field is an intermediate field.
5. Fixed fields can help determine whether an extension is separable or inseparable, with implications for the properties of roots of polynomials.

Review Questions

• How does understanding the fixed field help to establish the connection between Galois groups and field extensions?
  • The fixed field provides a vital link between Galois groups and their corresponding field extensions. By examining which elements remain unchanged under the action of the Galois group, we can identify subfields and understand how they relate to both the structure of the group and the properties of the extension. This relationship allows us to explore how different automorphisms interact with various elements and ultimately leads us to insights about solvability and normality within extensions.
• In what way does a normal extension relate to its fixed field, particularly regarding splitting polynomials?
  • In a normal extension, every irreducible polynomial that has at least one root in the extension must split completely within it. This property is directly tied to its fixed field because any automorphism from the Galois group will map roots of these polynomials into each other without introducing new roots outside this extension. Thus, analyzing how these roots behave under automorphisms helps us understand not just the fixed field but also why normal extensions maintain such splitting properties.
• Evaluate how fixed fields can impact our understanding of Frobenius automorphisms in relation to Artin symbols and their applications.
  • Fixed fields play a crucial role when examining Frobenius automorphisms, especially in number theory. In this context, they help identify how specific primes behave when viewed through the lens of Artin symbols. By linking Frobenius automorphisms to fixed fields, we can determine which elements remain unchanged when acting on roots or residues, leading to significant insights regarding local behavior in number fields and offering tools for solving problems related to class numbers and reciprocity laws.

Related terms

Galois group: The group of all automorphisms of a field extension that fix the base field, providing insights into the symmetries of the extension.

normal extension: A field extension where every irreducible polynomial in the base field that has at least one root in the extension splits completely in that extension.

splitting field: The smallest field extension in which a given polynomial splits into linear factors, often used to understand how roots behave under different automorphisms.