Timeline for Is there an analogue of finite fields for products of two prime powers?
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| Jan 14, 2012 at 16:25 | comment | added | Alan Haynes | @Goldstern No I don't. In retrospect there is probably no way to ask the question in a way that will solicit the kind of answer I am looking for, without knowing the answer in advance. I guess I was hoping that maybe someone out there had some sort of less direct construction that would give a class of objects with the desired property. Unfortunately that's all I can say to clarify, sorry it is so subjective. | |
| Jan 14, 2012 at 9:10 | comment | added | Goldstern | @AH: I agree that my description is not intrinsically different from saying that $n$ is a product of two prime powers. But do you have a mathematical definition of this concept of "intrinsically different"? | |
| Jan 12, 2012 at 18:40 | comment | added | Alan Haynes | @Goldstern In any case thank you for the answer, it is worthwhile to know that there is not some really ingenious model out there that springs to mind. I am tempted to mark your answer as correct, but who knows- maybe someone else knows of a class of objects from some other setting which has the property I am asking about. | |
| Jan 12, 2012 at 16:14 | comment | added | Emil Jeřábek | There is a very nice characterization of sets that are (first-order) spectra: exactly those that are computable in NE (nondeterministic exponential time), where $n$ is given in binary. | |
| Jan 12, 2012 at 16:02 | comment | added | Alan Haynes | @Goldstern Of course, it is a model theory problem. It sounds like what you are saying is that you can't think of a model for the set described above, besides the collection of pairs of finite fields with different characteristic- neither can I. The "product of two fields" is a misleading name- it's not a field, and however you define it, it's not intrinsically different than saying the number is a product of two prime powers. Felix's idempotent description is closer to the kind of model I am looking for, but still depends too intrinsically on the prime factorization. | |
| Jan 12, 2012 at 15:53 | comment | added | Andrés E. Caicedo | The paper also mentions a survey, "Fifty years of the spectrum problem", Durand, Jones, Makowsky, More. A preprint can be found here: www.diku.dk/hjemmesider/ansatte/neil/SpectraSubmitted.pdf | |
| Jan 12, 2012 at 15:51 | comment | added | Andrés E. Caicedo | As Martin says there is still much we do not know about spectra. A very recent paper on the subject is "Spectra and systems of equations", by Bell, Burris and Yeats, in "Model theoretic methods in finite combinatorics", AMS, Contemporary mathematics, vol 558 (2011), 43-96. | |
| Jan 12, 2012 at 14:53 | history | answered | Goldstern | CC BY-SA 3.0 |