automorfisen

Automorfism is a concept in abstract algebra and is part of the theory of finite fields and Galois theory. It specifically refers to an isomorphism from a field extension to itself. In other words, an automorfism is an isomorphism between a field and itself.

Automorfisms can play a significant role in understanding the structure of a field extension. They are crucial in the study of Galois groups and can be used to classify polynomials. An automorfism can also be seen as a permutation of the roots of a polynomial, which helps in studying the symmetries of polynomial equations.

Automorfisms can be obtained as the composition of the Frobenius automorphism with another endomorphism. In a finite field, every element can be seen as an automorfism of the field, known as the Frobenius automorphism. This particular automorfism is defined as raising the elements to a certain power.

In fact, the set of automorfisms of a Galois extension forms a group under composition. This group is known as the Galois group of the extension. The study of Galois groups has far-reaching implications in various areas of mathematics including algebraic number theory and field theory.

The term automorfism is derived from the Greek words "auto" meaning self and "morphism" meaning form. This etymology reflects the idea that an automorfism is a mapping from a field to itself. Automorphic functions and automorphic forms in number theory are related to this concept, but the terms generally refer to different mathematical objects.

The theory of automorfisms has numerous applications and has driven significant developments in abstract algebra and its related fields.