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I am new to the p-adic numbers (and also not too familiar with fields other than the reals, the rationals and $\mathbb{Z}/p$ ($p$ prime)). I want to know if the p-adic fields $\mathbb{Q}_p, \mathbb{Z}_p$ are characteristic 0 fields, or what are their characteristics? Actually, is $\mathbb{Z}_p$ a field at all?

(note that the only definition i have seen and somewhat understood so far of the p-adic number fields is that they are extensions of $\mathbb{Q}, \mathbb{Z}$ respectively (using Cauchy seq wrt the p-adic metric)).

Also what is the relationship between the p-adic numbers and characteristic $p$ fields?

Watson
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vkan
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    $\mathbb{Z}_p$ is not a field. Indeed $p$ has no inverse in it. And $\mathbb{Q}_p$ is of characterisitic 0. It contains $\mathbb{Z}$ as a subring. – Ahr Jun 05 '18 at 12:29
  • Hi Watson, thanks for your answer...i need to think about it and look up some terms/theory in your answer and will let you know if i understand it when ive done that. – vkan Jun 05 '18 at 14:10
  • Dear @vkan, did you look up the necessary theory? If so, then you may accept one of the answers ;-) Otherwise, please tell me what I should explain in more detail. – Watson Oct 29 '18 at 13:26

2 Answers2

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The ring $\Bbb Z_p$ is not a field, since $p$ is not invertible (its valuation is $1$, while the valuation of any unit is $0$). The fraction field is $\Bbb Q_p$, it is an infinite extension of $\Bbb Q$ (it is a completion of $\Bbb Q$), so it has characteristic $0$.

We have $\Bbb Z_p \cong \Bbb Z[[t]] / (t-p)$, but $\Bbb Z_p$ is not isomorphic to $\Bbb F_p[[t]]$. Moreover, $\Bbb Z_p / p^n \Bbb Z_p \cong \Bbb Z / p^n \Bbb Z$ for any prime $p$ and any $n \geq 0$. This gives you some relations with the rings of characteristic $p$.

(Notice that there are local fields of characteristic $p$, namely the finite extensions of the field of Laurent series $\Bbb F_q(\!(t)\!)$, where $q$ being a power of $p$).

Watson
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Hint: if it contains a copy of $\mathbb{Z}$ (as $\mathbb{Z}_p$ and $\mathbb{Q}_p$ do), it has characteristic $0$.

($\mathbb{Z}_p$ isn't a field, though. It's more properly called a ring.)

what is the relationship between the p-adic numbers and characteristic p fields?

For example, $\mathbb{Z}_p$ is a local ring, and its residue field has characteristic $p$.

Billy
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