Definition:Valued Field of P-adic Numbers

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Let $p$ be any prime number.

Let $\Q_p$ be the field of $p$-adic numbers.

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on $\Q_p$.

The valued field of p-adic numbers is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

Field of P-adic Numbers

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$.

Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$.

The field of $p$-adic numbers is the field $\Q_p$.

For any Cauchy sequence $\sequence{x_n}$ in $\struct{\Q, \norm {\,\cdot\,}_p}$, let $\eqclass{x_n}{}$ denote the left coset of $\sequence{x_n}$ in $\Q_p$.

$p$-adic Number

Each left coset $\eqclass{x_n}{}$ in $\Q_p$ is called a $p$-adic number.

P-adic Norm on P-adic Numbers

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:

$\ds \forall \eqclass{x_n}{} \in \Q_p: \norm {\eqclass{x_n}{} }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the $p$-adic norm on $\Q_p$.

Also see

  • Results about $p$-adic numbers can be found here.