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I looked online but I can't find any reasoning as to why the p-adic integers are called integers. Are there certain properties that $\mathbb{Z}_p$ shares with $\mathbb{Z}$, and is there sufficeint overlap to really associate the name integers with $\mathbb{Z}_p$? Or is this just a name that was used and stuck?

Understanding the similarity between (or lack thereof) will help with some of the intuition, or at least I hope it will. Right now I am finding things a bit confusing because $\mathbb{Z}_p$ emphatically contains things that are not integers.

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    $\begingroup$ Are you bothered by the name "Gaussian integers" for $\mathbf Z[i]$? That contains things that are not integers too. Anyway, the $p$-adic integers are a completion of the integers. $\endgroup$
    – KCd
    Oct 9, 2019 at 12:02
  • $\begingroup$ Sandstar how would you define $\Bbb{Z}_p$ ? $\endgroup$
    – reuns
    Oct 9, 2019 at 12:06
  • $\begingroup$ $\mathbb{Z}_p = \{x\in \mathbb{Q}_p : |x|_p\leq1\}$ $\endgroup$
    – Sandstar
    Oct 9, 2019 at 12:22

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$\Bbb Z_p$ emphatically contains things that are integers as well. Loosely speaking, it's the integers base $p$, but we're allowing the "digits" to extend infinitely far to the left. In contrast, for real numbers, we allow an infinite extension to the right (after the decimal point).

There is a subset of $\Bbb Z_p$ of elements whose expansion only continues finitely far to the left. This subset is a ring, and it's isomorphic to the integers.

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$\mathbb{Q}$ is the quotient field of the ring of (rational) integers $\mathbb{Z}$

And $\mathbb{Q}_p$ is the quotient field of the ring of p-adic integers $\mathbb{Z}_p$

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