nLab p-adic complex number

Idea

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

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.