feltstruktur

Feltstruktur, or field structure in mathematics, describes the algebraic framework of a set equipped with two binary operations, addition and multiplication, that satisfy the axioms of a field. A field is a commutative ring with unity in which every nonzero element has a multiplicative inverse. Equivalently, (F,+) is an abelian group with identity 0, (F\{0},×) is an abelian group with identity 1, and multiplication distributes over addition.

Examples include the rational numbers Q, the real numbers R, and the complex numbers C, which share the same field axioms. Finite fields, such as GF(p) for a prime p and their extensions GF(p^n), are central in number theory and coding theory. The characteristic of a field is either 0 (as in Q, R, C) or a prime p (as in GF(p^n)). A prime field is the smallest subfield isomorphic to either Z or Z/pZ.

Substructures and maps: a subset of a field that is closed under the two operations and contains 0 and 1 that is itself a field is a subfield. Field extensions L/K are larger fields containing K; the degree [L:K] measures their size. Morphisms that preserve both operations are called field homomorphisms; isomorphisms indicate structural equivalence, and automorphisms form the symmetry group of a field.

Applications: field structure is fundamental in algebra, number theory, and algebraic geometry; it underpins the study of polynomials, solvability by radicals, and Galois theory. Not every ring is a field; the requirement of multiplicative inverses for all nonzero elements is essential to the field notion.