# History of the notation $\mathbb Z_n$

This question was motivated by Martin's comment in Free $\mathbb{Z}_2$-actions match at some point

When was the notation $\mathbb Z_n$ introduced for $n$-adic integers and by whom? When was it introduced for integers modulo $n$ and by whom?

I tried searching on Google without much luck.

Added. Martin brings up the related point when was the subscript notation $A_f$ introduced for localizing a ring at the monoid generated by $f$ (which is a third possible interpretation of $\mathbb Z_n$)?

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It is pretty common to denote the cyclic group of order $n$ by $C_n\dots$ – Igor Rivin Nov 25 '11 at 22:11
I looked in Hensel's original book Theorie der Algebraischen Zahlen and he writes ${\mathbf Q}_p$ as $K(p)$ and, as far as I could tell, he has no specific notation for $p$-adic integers. He just refers to them when needed with the words "$p$-adic integers" (in German). In Hasse's book Number Theory the $p$-adic numbers are $P_p$ and the $p$-adic integers are $\Gamma_p$. – KConrad Nov 25 '11 at 23:52
Borevich-Shafarevich's book "Number Theory" (1960s) denotes the $p$-adic integers as $O_p$, $p$-adic numbers as $R_p$ (since they write $R$ for the rational numbers). Dwork's paper on rationality of the zeta-function (Amer. J. Math 1960) denotes $p$-adic numbers as $Q'$ and $p$-adic integers as ${\mathfrak O}'$. Lubin's paper on formal groups (Ann. Math. 80, 1964) uses ${\mathbf Z}_p$ for the $p$-adic integers and ${\mathbf Q}_p$ for the $p$-adic numbers. Someone should look at the first edition of Serre's Corps Locaux (1962). – KConrad Nov 26 '11 at 0:34
@Benjamin Steinberg: I think it'd be more accurate to say $C_n$ is a notation for the cyclic group abstractly (using multiplication for the group law notation, as usual in abstract groups) rather than the cyclic group multiplicatively, because many people might want to write that as $\mu_n$. – KConrad Nov 26 '11 at 1:38
I have the 1968 edition of Corps Locaux, which is a photographic reproduction of the first edition. Serre Uses ${\mathbf{Q}}_p$ for the $p$-adic numbers, and ${\mathbf{Z}}_p$ for their integers. I'm pretty sure that I learned this notation from Lang, in course I took with him at Columbia, in 1956-57. – Lubin Nov 26 '11 at 5:56

I fished around in Google scholar and found so many examples that I don't feel like listing any of the links. Nonetheless, a clear picture emerges of an answer that I found a bit surprising: The notation $\mathbb{Z}_p$ for the $p$-adic integers evolved in three separate parts. I should also explain that the real science of etymology is about the evolution of words or notation, not just "when did it first happen".

The subscript notation not only for the $p$-adic integers, but more generally for $p$-adic completions, already appears in several papers in the 1930s and 1940s. For instance Carl Ludwig Siegel says in 1941, "$R$ is the field of rational numbers, $R_p$ the field of $p$-adic numbers, where $p$ denotes any prime number, $R_\infty$ the field of real numbers; moreover $J$ is the ring of integral numbers and $J_p$ the ring of $p$-adic integers". Of course, no one would use this notation today!

The use of $Z$ for the integers has a semi-separate history. I even found an old paper, but more recent than this one by Siegel, that used $Z$ for the integers but $R$ for the $p$-adic integers, with no subscript.

Generally the notation for $p$-adic integers and $p$-adic numbers standardized at $Z_p$ and $Q_p$ in the 1950s. Quite possibly Bourbaki, Algebra, deserves credit for standardizing $Z$ and $Q$ for integers and rationals.

Blackboard bold notation ($\mathbb{Z}$ and $\mathbb{Q}$) came last, at least in print. Despite its name, it's no longer obvious to me that blackboard bold actually first came from blackboards or from typewriters. It's sometimes also credited to Bourbaki, but this seems to be wrong. There is a historical account by Lee Rudolph (in comp.text.tex) that credits certain typewriter models in the 1960s for producing blackboard bold typography for the integers, etc. If that is where it started, then the notation seemed to catch on fairly quickly, although there were holdouts that used ordinary bold for decades after that. (But, before blackboard bold was fashionable, it wasn't even standard to make the set of integers bold $\mathbf{Z}$ instead of just $Z$.)

As an aside, the collision of notation between the $p$-adic integers and the integers mod $p$ is unfortunate. I really prefer to write $\mathbb{Z}/n$ for the integers mod $n$, because it is then written exactly as it reads. Also, partly since it is such a commonly used object, I see no need for extra parentheses, or an extra $\mathbb{Z}$, and certainly just using an $n$ subscript is bad. I'm optimistic that this notation is the way of the future and it would be an interesting separate question in history of notation.

(Sorry, I didn't see the entire string of comments before I wrote all of this. The comments make most of these remarks, but it seems useful to combine them into one historical summary.)

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Did you see anything indicating when some for of Z subscript n appeared (blackboard or not) for integers mod n? The comments above seemed to have already landed the subscript notation for completions at the door of Hasse in the thirties. – Benjamin Steinberg Nov 27 '11 at 2:30
One should check what notation, if any, van der Waerden used in Moderne Algebra (1930) for the integers mod $n$ since whatever convention he chose influenced the next generations in algebra but not necessarily other parts of mathematics. (Incidentally, in his book Number Theory (1950), Hasse wrote $Z_\infty$ for the complex numbers.) – KConrad Nov 27 '11 at 2:46
I found a paper by one John Moore (1957) that uses $Z_n$ for integers mod $n$ and $Z^p$ for $p$-adic integers. So that's one solution, apparently obsolete, for the collision-of-notation problem. Now I see a couple of other papers in the 1950s. It seems that the standard of $Z$ for the integers, which appeared in that period, swept in the collision of notation immediately and unintentionally. Nakayama (1957) writes $Z(p)$ for $\mathbb{Z}/p$, integers mod $p$, while Floyd (1951) writes $I_p$ for the same. – Greg Kuperberg Nov 27 '11 at 3:01
Greg: It's okay not to give links, but could you indicate the journal and not just the year? It makes it easier to check things. – KConrad Nov 27 '11 at 3:31
– Greg Kuperberg Nov 27 '11 at 3:39