# Number Fields (Universitext) 2nd Edition

The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements.

## Number Fields (Universitext) 2nd Edition

Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A, and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms above), and I is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,

A field is called a prime field if it has no proper (i.e., strictly smaller) subfields. Any field F contains a prime field. If the characteristic of F is p (a prime number), the prime field is isomorphic to the finite field Fp introduced below. Otherwise the prime field is isomorphic to Q.[14]

The first clear definition of an abstract field is due to Weber (1893).[24] In particular, Heinrich Martin Weber's notion included the field Fp. Giuseppe Veronese (1891) studied the field of formal power series, which led Hensel (1904) to introduce the field of p-adic numbers. Steinitz (1910) synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. Artin & Schreier (1927) linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties.[25] Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.

Any field F has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as the algebraic closure and denoted F. For example, the algebraic closure Q of Q is called the field of algebraic numbers. The field F is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice.[34] In this regard, the algebraic closure of Fq, is exceptionally simple. It is the union of the finite fields containing Fq (the ones of order qn). For any algebraically closed field F of characteristic 0, the algebraic closure of the field F((t)) of Laurent series is the field of Puiseux series, obtained by adjoining roots of t.[35]

Basic invariants of a field F include the characteristic and the transcendence degree of F over its prime field. The latter is defined as the maximal number of elements in F that are algebraically independent over the prime field. Two algebraically closed fields E and F are isomorphic precisely if these two data agree.[49] This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic. For example, Qp, Cp and C are isomorphic (but not isomorphic as topological fields).

Global fields are in the limelight in algebraic number theory and arithmetic geometry.They are, by definition, number fields (finite extensions of Q) or function fields over Fq (finite extensions of Fq(t)). As for local fields, these two types of fields share several similar features, even though they are of characteristic 0 and positive characteristic, respectively. This function field analogy can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the Riemann hypothesis concerning the zeros of the Riemann zeta function (open as of 2017) can be regarded as being parallel to the Weil conjectures (proven in 1974 by Pierre Deligne).

The study of algebraic number fields is arguably the backbone of all of number theory. One generally subdivides number theory into four subdisciplines, namely, elementary number theory, geometric number theory, analytic number theory, and algebraic number theory; algebraic number fields very properly belong to the latter subdiscipline. Indeed, the study of number fields is essentially coextensive with algebraic number theory, at least if one allows the inclusion of local fields into the discussion. 041b061a72