Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1
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Theorem
Let $p$ be a prime number.
Let $\Z_p$ be the $p$-adic integers.
The set of integers $\Z$ is a subring of $\Z_p$.
Proof
Let $\Z_{\paren p}$ be the valuation ring induced by $\norm {\,\cdot\,}_p$ on $\Q$.
By Integers form Subring of Valuation Ring of P-adic Norm on Rationals then:
- $\Z$ is a subring of $\Z_{\paren p}$
By Valuation Ring of P-adic Norm is Subring of P-adic Integers then:
- $\Z_{\paren p}$ is a subring of $\Z_p$
The result follows.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.3$ Exploring $\Q_p$, Proposition $3.3.4$