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If x is an element of a field K and n is a positive integer, we have both a symbol and a name for a root of the polynomial t^n - x = 0: we denote it by x^{1/n} and call it an nth root of x.

Of course nth roots play a vital role in field theory, e.g. in the characterization of solvable extensions in characteristic 0. However, in characteristic p > 0, the extraction of a p-power root is a much different business: it gives rise to purely inseparable extensions, not composition factors of solvable Galois extensions.

To repair the characterization of solvable extensions in characteristic p as those being attainable as a tower of "radical" extensions, one needs to include the operation of taking roots of Artin-Schreier polynomials: t^q - t - x = 0, for q = p^a a power of the characteristic.

Finally my question: do we have a name for an element t solving the equation t^q - t = x and/or a special notation for it? I do not know one. Similarly, whereas classically we often speak of x as being "an nth power", in this case I find myself writing "x is in the image of the Artin-Schreier isogeny \rho". Is there something better than this?

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Can't you just call them Artin-Schreier roots? I mean, it's not like that could mean anything else. –  Harry Gindi Dec 8 '09 at 6:53
    
I feel that "p^a-Artin Schreier root" is a bit wordy, but it may be the best available. –  Pete L. Clark Dec 8 '09 at 7:36
    
Too mundane for an answer, but just stumbled across the name 2-root here: en.wikipedia.org/wiki/Quadratic_formula#Characteristic_2 –  Cam McLeman Sep 7 '10 at 3:23

2 Answers 2

up vote 3 down vote accepted

Google Scholar finds three papers with the phrase "Artin-Schreier root" or "Artin-Schreier roots" (with quotes). The papers are by Jing Yu, Thomas Scanlon, and Spencer Bloch + Helene Esnault. This is not all that many, but maybe enough for some kind of standard. Various people, probably enough to call it standard, also use the notation $\wp(x)$ (Weirstrass p, or \wp in amslatex) for the Artin-Schreier polynomial. So you could use the notation $\wp^{-1}(x)$ for an Artin-Schreier root of $x$. (The notation is unambiguous, because which polynomial you take is determined by the characteristic of the field containing $x$.) The notation is often used to mean the first Artin-Schreier polynomial with $q=p$, which is unambiguous but only in a lame sense. (As Pete was too polite to emphasize in his comment, I should have read the question more carefully.)


If you're bothered by the length of the phrase "Artin-Schreier root" and you want to be a real joker with terminology, you could call it an "asroot". (The current use of this word is a Unix command which means to execute something as root user. You could pronounce it the same except with the emphasis on the first syllable.) Why not? Gromov got away with the barbarous term "a-T-menable". And who came up with "rng" and "rig"?

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Just a comment on the parenthetical remark: that's not true, since we have the p^a-Artin-Schreier isogeny for all positive integers a. But perhaps \wp^{-1}_q(x) (or \rho^{-1}_q(x)) is a good notation... –  Pete L. Clark Dec 8 '09 at 7:34
    
The notation $\wp_q(x)$ and $\wp^{-1}_q(x)$ seems naturally consistent with the notation such $\mathbb{F}_q$. While you're at it, why not $q$-Artin-Schreier root. Also the Google searches did reveal some cumbersome descriptions; you may have the chance to innovate with notation. –  Greg Kuperberg Dec 8 '09 at 8:05
    
Yes, there's some terrible terminology out there -- that's not exactly motivational! Asroot is, of course, ridiculous. So are "rng" and "rig": I use the former myself, but only to drive home the point that a ring without a multiplicative identity is missing something very important. "rig" is especially bad, since a much better term already exists: semiring. –  Pete L. Clark Dec 28 '09 at 6:00
    
I think that silly terminology isn't always bad terminology. The terminology "pin group" is blue as well as silly, but it's still a good name. Maybe silly and bad is worse than sober and bad. I suspect that you're free to be as silly or sober as you want in this case. en.wikipedia.org/wiki/Pin_group –  Greg Kuperberg Dec 28 '09 at 6:21

I have been using the notation $\wp^{-1}(a)$ for a root of $T^p-T-a$ (in characteristic $p$), which I borrowed from Bourbaki, but I have also used $\wp_2^{-1}(a)$ and $\wp_3^{-1}(a)$ if the characteristic had to be mentioned explicitly.

This morning I came across an original notation for $\wp_p^{-1}(a)$ in a paper by Davenport and Hasse in the Journal für die reine und angewandte Mathematik, 172 (1935). I cannot display it here because Knuth was not aware of it when he created TeX, so all I can do is to provide a link to their article. The notation appears on the last line of the first page (p.151) and it is defined in the first line of the second page (p.152).

Clearly, the reason behind their notation is its similarity to $\root p\of{}$. It is a less angular and more curvaceous version of the radical, with the loop closing upon itself.

I wish the people responsible for TeX and Unicode will give a second life to this little gem.

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You could use \AchtBR from the harmony.sty package :) –  Mariano Suárez-Alvarez Dec 20 '11 at 4:49
    
Nice suggestion (ctan.mirrorcatalogs.com/macros/latex/contrib/harmony/…), Mariano. I wonder whether editors and referees will allow me to get away with it. Hasse could do it because he was one of the editors of Crelle at the time. –  Chandan Singh Dalawat Dec 20 '11 at 5:16

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