Complex p-adic numbers

Description

Let $p$ be a prime and let $\Q_p$ be the field of p-adic numbers with the $p$ -adic absolute value $\abs{\cdot}_p$ . Let $\overline{\Q}_p$ be an algebraic closure of $\Q_p$ . Because $\Q_p$ is complete, $\abs{\cdot}_p$ extends in exactly one way to an absolute value on $\overline{\Q}_p$ , and this extension is a non-Archimedean absolute value. The field $\overline{\Q}_p$ is not complete for it. The field $\C_p$ of complex $p$ -adic numbers is the completion of $\overline{\Q}_p$ for this absolute value.

The field $\C_p$ is an algebraically closed field and it is complete. The algebraic closedness follows from Krasner's lemma. One approximates a monic polynomial over $\C_p$ by monic polynomials over $\overline{\Q}_p$ , whose roots lie in $\overline{\Q}_p$ . The roots of these approximations can be chosen to form a Cauchy sequence, and its limit in $\C_p$ is a root of the original polynomial.

Remarks

The set of values $\abs{x}_p$ for $x \in \C_p^\times$ is $p^{\Q} = \set{p^r : r \in \Q}$ , the same as for $\overline{\Q}_p$ . The residue field of $\C_p$ is an algebraic closure of the finite field $\F_p$ . The field $\C_p$ plays for $\Q_p$ the role that $\C$ plays for $\R$ , but $\C_p$ has infinite degree over $\Q_p$ .

Tag	`FKF9YGBKHF`
URI	`m2://FKF9YGBKHF`
Kind	object
Writing	W0 stub
MSC	12J25, 11S80
Instance of	Algebraically closed field
Uses	p-adic number, Algebraic closure, Completion of a valued field, Non-Archimedean absolute value
Changed	2 November 2026

Tag

FKF9YGBKHF

URI

m2://FKF9YGBKHF

Kind

object

Writing

W0 stub

MSC

12J25, 11S80

Instance of

Algebraically closed field

Uses

p-adic number, Algebraic closure, Completion of a valued field, Non-Archimedean absolute value

Changed

2 November 2026

Complex p-adic numbers

Description

Remarks

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