There seem to be two conflicting definitions for *p-adic valuation* in the literature.


Firstly, for any non-zero integer n, we have $\nu=\nu\_p(n)$ is the greatest non-negative integer such that $p^\nu$ divides $n$. Secondly, we have $|n|\_p$ which is defined as $1/p^\nu$. [These definitions can be extended to the rationals.]


$\nu$ is defined as the p-adic valuation in Khrennikov, Nilson, *P-adic deterministic and random dynamics* (for example) and $|\cdot|\_p$ is defined as the p-adic valuation in Khrennikov, *P-adic and group valued probabilities*, in Harmonic, wavelet and p-adic analysis (for example).



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> Question: Is there a preferred definition for p-adic valuation?
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What is the p-adic valuation of a number?
I will explain what's going on. We call $\lvert x\rvert\_p$ the $p$-adic absolute value of $x$ and $v\_p(x)$ the $p$-adic valuation of $x$. The distinction that is made by the two terms "absolute value" and "valuation" is completely standard… in English. However, Khrennikov is originally from Russia and in Russian there is one term for both concepts (нормирование = normirovanie, with stress on the second syllable -- I am not making that up, but see comments below this answer about stress on derived words in Russian, like verbs becoming nouns or nouns becoming adjectives). There is a term "absolute value" in Russian, but it is not an abstract concept; it refers only to the usual absolute value on the real or complex numbers (and quaternions?). This is perhaps why Khrennikov is using the term "valuation" incorrectly to refer to an absolute value function.


(I'm giving a course in Moscow this semester and I found this point *frustrating* when I was preparing my initial lectures. In different books I found the same word used for an absolute value and for a valuation and couldn't find the term that exclusively means absolute value. Eventually I determined there isn't one; you just know by context what meaning is intended. Native speakers are welcome to correct me here.)


UPDATE (3 years later): I learned from a student in St. Petersburg that the mathematicians there use separate terms for an absolute value $\lvert\cdot\rvert$ on a field and its corresponding valuation $v$: they call $\lvert x\rvert$ the norm (норма) of $x$ and $v(x)$ the exponent (показатель) of $x$.


UPDATE (11 years later): Consistent with Laurent's answer, the [Russian Wikipedia page for absolute value](https://ru.wikipedia.org/wiki/%D0%90%D0%B1%D1%81%D0%BE%D0%BB%D1%8E%D1%82%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0) (find the English one and then click on Русский) refers to an absolute value as нормирование, a word I mentioned in the first paragraph above, and a valuation as экспоненциальное нормиорование, where the word in front of нормирование is eksponentsialnoe = exponential.


