Zahlkörper

A Zahlkörper, or number field in English, is a fundamental concept in abstract algebra and number theory. It is defined as a finite algebraic extension of the field of rational numbers, denoted by Q. In simpler terms, a number field is a set of numbers that contains all rational numbers and also satisfies certain properties related to roots of polynomials.

More formally, a number field K is a field such that there exists a field embedding of K into the complex numbers C, and the field K can be viewed as a vector space over Q of finite dimension. This dimension is called the degree of the number field. For example, the field of Gaussian integers Q(i), which consists of numbers of the form a + bi where a and b are rational numbers and i is the imaginary unit, is a number field of degree 2 over Q.

The study of number fields is central to algebraic number theory. It involves examining the ring of integers within a number field, which are the elements that are roots of monic polynomials with integer coefficients. Properties like unique factorization of ideals in this ring are crucial for understanding the arithmetic of the number field. Number fields provide a framework for generalizing concepts from elementary number theory, such as prime numbers and divisibility, to a broader algebraic setting.