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?



Notation/name for "Artin-Schreier roots"?
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.)




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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"?


