nLab
p-adic complex number

Context

Arithmetic geometry

Algebra

Contents

Idea

For pp a prime number, the field of complex pp-adic numbers p\mathbb{C}_p is to the p-adic numbers p\mathbb{Q}_p as the complex numbers \mathbb{C} are to the real numbers.

Definition

First observe that the ordinary complex numbers \mathbb{C} may be characterized as follows:

the standard absolute value (norm) on the rational numbers \mathbb{Q} uniquely extends to an algebraic closure ¯\bar \mathbb{Q}, and the completion is the complex numbers.

In direct analogy with this:

for pp a prime number and p\mathbb{Q}_p the corresponding non-archimedean field of p-adic rational numbers, then the completion of any algebraic closure ¯ p\bar \mathbb{Q}_p is the field of complex pp-adic numbers p\mathbb{C}_p.

Notice that the completion of the algebraic closure of a normed field is still algebraically closed (Bosch-Guntzer-Remmert 84, prop. 3.4.1.3). See also at normed field – relation to algebraic closure.

References

  • L. C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.

  • PlanetMath, complex p-adic numbers

  • S. Bosch, U. Guntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984.

Last revised on February 18, 2017 at 00:18:33. See the history of this page for a list of all contributions to it.