Category:Equivalence of Definitions of P-adic Norms
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This category contains pages concerning Equivalence of Definitions of P-adic Norms:
Let $p \in \N$ be a prime.
Let $\Q$ denote the rational numbers.
The following definitions of the concept of $p$-adic norm on $\Q$ are equivalent:
Definition 1
Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.
The $p$-adic norm on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:
- $\forall q \in \Q: \norm q_p := \begin{cases}
0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$
Definition 2
The $p$-adic norm on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:
- $\forall r \in \Q: \norm r_p = \begin {cases} 0 & : r = 0 \\ \dfrac 1 {p^k} & : r = p^k \dfrac m n: k, m, n \in \Z, p \nmid m, n \end {cases}$
Pages in category "Equivalence of Definitions of P-adic Norms"
The following 2 pages are in this category, out of 2 total.