Complete field

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In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. A field supports the elementary operations of addition, subtraction, multiplication, and division, while a metric represents the distance between two points in the set. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers).

Contents

Definitions

Field

A field is a set with binary operations and (called addition and multiplication, respectively), along with elements and such that for all , the following relations hold: [1]

  1. has a solution
  2. and
  3. has a solution for

Complete metric

A metric on a set is a function , that is, it takes two points in and sends them to a non-negative real number, such that the following relations hold for all : [2]

  1. if and only if

A sequence in the space is Cauchy with respect to this metric if for all there exists an such that for all we have , and a metric is then complete if every Cauchy sequence in the metric space converges, that is, there is some where for all there exists an such that for all we have . Every convergent sequence is Cauchy, however the converse does not hold in general. [2] [3]

Constructions

Real and complex numbers

The real numbers are the field with the standard Euclidean metric , and this measure is complete. [2] Extending the reals by adding the imaginary number satisfying gives the field , which is also a complete field. [3]

p-adic

The p-adic numbers are constructed from by using the p-adic absolute value

where Then using the factorization where does not divide its valuation is the integer . The completion of by is the complete field called the p-adic numbers. This is a case where the field is not algebraically closed. Typically, the process is to take the separable closure and then complete it again. This field is usually denoted [4]

References

  1. Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (Third ed.). Boston, MA: Brooks/Cole, Cengage Learning. pp. 44, 49. ISBN   978-1-111-56962-4.
  2. 1 2 3 Folland, Gerald B. (1999). Real analysis: modern techniques and their applications (2nd ed.). Chichester Weinheim [etc.]: New York J. Wiley & sons. pp. 13–14. ISBN   0-471-31716-0.
  3. 1 2 Rudin, Walter (2008). Principles of mathematical analysis (3., [Nachdr.] ed.). New York: McGraw-Hill. pp. 47, 52–54. ISBN   978-0-07-054235-8.
  4. Koblitz, Neal. (1984). P-adic Numbers, p-adic Analysis, and Zeta-Functions (Second ed.). New York, NY: Springer New York. pp. 52–75. ISBN   978-1-4612-1112-9. OCLC   853269675.

See also