Automorphism

Definition

An automorphism is an isomorphism from a mathematical object to itself, preserving the structure of that object. This concept is crucial in understanding symmetries and transformations within algebraic structures, especially when considering extensions, fields, and polynomials, as it reveals intrinsic properties that remain unchanged under these mappings.

5 Must Know Facts For Your Next Test

1. Automorphisms play a central role in Galois theory, particularly in connecting field extensions and the roots of polynomials through their symmetry.
2. In the context of Galois extensions, the set of all automorphisms forms a group, known as the Galois group, which helps in understanding how different extensions relate to each other.
3. The identity automorphism is the simplest example, mapping every element to itself and acting as a neutral element in the structure of automorphisms.
4. Automorphisms can provide insights into the separability of polynomials; if every root of a polynomial can be transformed into another root via an automorphism, the polynomial is likely separable.
5. The fixed field of an automorphism can reveal key information about the underlying algebraic structure and its subfields.

Review Questions

• How does the concept of automorphism help us understand the structure of Galois extensions?
  • Automorphisms allow us to see how different field extensions relate through symmetries. In Galois theory, automorphisms that fix the base field form a group called the Galois group, which captures essential properties about the roots of polynomials. This relationship between automorphisms and field extensions helps us analyze solvability and other properties of polynomial equations.
• What role do automorphisms play in determining whether a polynomial is separable or not?
  • Automorphisms can show whether all roots of a polynomial can be interchanged without introducing any new roots. If there exists an automorphism that maps each root to another root within the same field, this indicates that the polynomial is separable. The presence of repeated roots, however, implies that some automorphisms cannot perform this function, suggesting inseparability.
• Analyze how automorphisms relate to minimal polynomials and their degree in field extensions.
  • Automorphisms provide insight into minimal polynomials by illustrating how roots transform within field extensions. The degree of these minimal polynomials can often be connected to the number of distinct roots preserved by automorphisms. When an automorphism keeps a polynomial's structure intact, it hints at underlying symmetries that might simplify our understanding of its roots and their relationships within larger fields.

Related terms

Isomorphism: A bijective homomorphism between two algebraic structures that preserves operations and relations.

Galois Group: A group consisting of all automorphisms of a field extension that fix the base field, revealing the symmetry of the roots of polynomials.

Fixed Field: The subfield of elements in a field extension that remain unchanged under a given automorphism.