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).

Definitions

Field

A field is a set F with binary operations + and ⋅ (called addition and multiplication, respectively), along with elements 0 and 1 such that for all a,b,c∈F, the following relations hold:[1]

  1. a+(b+c)=(a+b)+c
  2. a+b=b+a
  3. a+0=a=0+a
  4. a+x=0 has a solution
  5. a(bc)=(ab)c
  6. ab=ba
  7. a(b+c)=ab+ac and (a+b)c=ac+bc
  8. a1=a=1a
  9. ax=1 has a solution for a≠0

Complete metric

A metric on a set F is a function d:F2→[0,∞), that is, it takes two points in F and sends them to a non-negative real number, such that the following relations hold for all x,y,z∈F:[2]

  1. d(x,y)=0 if and only if x=y
  2. d(x,y)=d(y,x)
  3. d(x,y)≤d(x,z)+d(z,y)

A sequence xn in the space is Cauchy with respect to this metric if for all ϵ>0 there exists an N∈ℕ such that for all n,m≥N we have d(xn,xm)<ϵ, and a metric is then complete if every Cauchy sequence in the metric space converges, that is, there is some x∈F where for all ϵ>0 there exists an N∈ℕ such that for all n≥N we have d(xn,x)<ϵ. 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 |x−y|, and this measure is complete.[2] Extending the reals by adding the imaginary number i satisfying i2=−1 gives the field ℂ, which is also a complete field.[3]

p-adic

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

vp(a/b)=vp(a)−vp(b)

where a,b∈ℤ. Then using the factorization a=pnc where p does not divide c, its valuation is the integer n. The completion of ℚ by vp is the complete field ℚp 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 ℂp.[4]

References

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  1. ^ Page Module:Citation/CS1/styles.css has no content.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. ^ a b c Page Module:Citation/CS1/styles.css has no content.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. ^ a b Page Module:Citation/CS1/styles.css has no content.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. ^ Page Module:Citation/CS1/styles.css has no content.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

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