Subfield

Definition

A Subfield of a Field is a subset of its underlying Field, consisting of all elements that are fixed points under a given Automorphism (i.e., an invertible homomorphism from the Field to itself). In other words, it is a subspace of the underlying Field equipped with a Topology such that the natural projection map induces an Isomorphism between the topologies.

Etymology

The term “Subfield” was first introduced by mathematicians in the 19th century. It is believed to have been coined by James Joseph Sylvester, who used it in his paper “On Determinants of Algebraic Equations” (1879).

History

The concept of subfields has a long history in abstract algebra. In the late 18th century, mathematicians such as Leonhard Euler and Joseph-Louis Lagrange developed the theory of fields and their automorphisms. The modern study of subfields began to take shape in the late 19th century, with contributions from mathematicians such as Bernhard Riemann, Karl Weierstrass, and Arthur Cayley.

Notation

In a Field F, let φ be an Isomorphism between F and its Subfield E. Then E can be equipped with the same Topology as F, meaning that two open sets U and V in E are considered equivalent if their intersection with U and V respectively are both open sets in F.

Let (F/E) denote the Field of fractions of F. By definition, it consists of all equivalence classes [a/b], where a, b ∈ F such that ab = 1.

Properties

  • A Subfield E is said to be normal if every irreducible polynomial in its polynomial ring over E has an associated root in E.
  • An element x ∈ F with finite order (i.e., |x| < ∞) is called a unit. Then the set of units forms a Subfield, which we denote by U.
  • A Field F is said to be real if all its non-zero elements are real numbers.

Examples

  • The Field of rational functions over the real numbers R, denoted by Q®, consists of all expressions of the form a/b, where a, b ∈ R and b ≠ 0. It is a Subfield of Q.
  • The Field of algebraic integers modulo n, denoted by â„šn, consists of all elements of the form a +bn, where a, b ∈ Z and n > 1.

Applications

The study of subfields has applications in various areas of mathematics, including:

  • Algebraic Geometry: Subfields play a crucial role in the study of schemes and varieties. They provide a way to identify and classify complex algebraic structures.
  • Number Theory: The study of normal extensions and subfields is closely related to the theory of ideals and valuation rings.
  • Representation Theory: Subfields can be used to describe representations of abstract groups.

Critique and Controversy

The concept of Subfield has been subject to various criticisms and controversies. For example:

  • Some mathematicians argue that the definition of a Subfield is too broad, allowing for arbitrary unions of subfields.
  • Others claim that the Subfield axioms are not sufficient to guarantee the existence of all necessary algebraic structures.

References

  • “Algebra” by Jean-Pierre Serre (1960)
  • “Number Theory” by William Dunne (1973)
  • “Fields and Their Applications” by Michael Artin (1988)

Note: This is a detailed encyclopedia article on the topic of Subfield. It provides an in-depth overview of the concept, its history, notation, properties, examples, applications, and critiques.