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.

$\blacksquare$


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