Timeline for Is there an analogue of finite fields for products of two prime powers?
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2012 at 15:50 | answer | added | felix | timeline score: 1 | |
| Jan 14, 2012 at 13:54 | history | edited | Goldstern |
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| Jan 13, 2012 at 2:07 | answer | added | Noam D. Elkies | timeline score: 4 | |
| Jan 12, 2012 at 22:01 | answer | added | Kevin O'Bryant | timeline score: 3 | |
| Jan 12, 2012 at 18:36 | comment | added | Alan Haynes | @quid Not at all- I think your comment was a good one, but I don't have an answer on hand. That is why I chose to reword the question. I do think that if we think about it we could come up with some answers to your question, but right now I am more interested in the question in the title. | |
| Jan 12, 2012 at 18:09 | comment | added | Alan Haynes | @Emil Jerabek I don't have a specific problem in mind at the moment. The motivation is that I have been working on some problems in finite fields and this question just came to mind- it is something that I have wondered about on and off for some time. | |
| Jan 12, 2012 at 18:02 | comment | added | user9072 | @AH: Sorry for the confusion. I was really just genuinely curious whether you have/had specific instances in mind where this characterization of prime powers is used (as a characterization of prime powers). I do not know one but know some things sufficiently close that it seemed conveivable there is one (the prime thing I mentioned, or it can also be useful to have nonneg ints characterized as sums of four powers of ints, while this perhaps not being the default way how Lagrange's 4-squares thm is thought about). Thus the question.. | |
| Jan 12, 2012 at 17:05 | comment | added | Emil Jeřábek | @AH: I think quid was not trying to argue with you. There is no controversy, it’s just that it would help understand what kind of answer you are looking for if you included some motivation in the question. What do you need the property $P(n)$ for? | |
| Jan 12, 2012 at 16:12 | comment | added | Alan Haynes | @quid Ok I'm not going to argue, I will reword the question to avoid controversy. | |
| Jan 12, 2012 at 16:10 | history | edited | Alan Haynes | CC BY-SA 3.0 |
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| Jan 12, 2012 at 15:12 | comment | added | user9072 | This is a bit tangential, but where is it extremely useful that prime powers are chatacterized in the way given in the answer? To make more precise what I mean, of course it is useful to have a sort of complete description of finite fields. And, I also would know why it is useful that say primes are chatacterized by the fact that Z/nZ is a field. But why this characterization of prime powers is useful I would not know. | |
| Jan 12, 2012 at 15:11 | comment | added | Simon Wadsley | Isn't the trouble with this products of two fields example that you have to ensure the fields don't have the same characteristic? I suppose you could add the condition that there is no field of that order. | |
| Jan 12, 2012 at 14:53 | answer | added | Goldstern | timeline score: 5 | |
| Jan 12, 2012 at 14:26 | comment | added | Matt Brin | So I lucked out. Should not try to think this early. On original, do the automorphism groups help? | |
| Jan 12, 2012 at 14:16 | comment | added | felix | (I should have written, "product of $m \ge 2$ distinct primes", i.e. these numbers are precisely the non-prime squarefree numbers. 1 is also included in this list.) | |
| Jan 12, 2012 at 14:13 | comment | added | felix | @Matt: but that only works for product of $m$ primes (and not just two), and not for product of two (or more) prime powers. | |
| Jan 12, 2012 at 13:53 | comment | added | Matt Brin | Every abelian group of order n is cyclic, but there is no field of order n? | |
| Jan 12, 2012 at 13:42 | comment | added | felix | (This also works for numbers with precisely $m$ prime factors: use such rings with precisely $2^m$ idempotent elements. And as with finite fields, such rings are unique up to isomorphism.) | |
| Jan 12, 2012 at 13:15 | comment | added | felix | You could classify products of two finite fields as finite commutative unitary rings which have precisely four idempotent elements and which are reduced (i.e. no nilpotent elements). Would a statement of $P(n)$ using this classification of products of two finite fields be acceptable? | |
| Jan 12, 2012 at 11:51 | comment | added | Alan Haynes | @Emil Jerabek Yes, that is what I was trying to say at the end of the question- I am looking for a characterization which doesn't just come from putting together two prime powers to form a new object. | |
| Jan 12, 2012 at 11:28 | history | edited | Alan Haynes | CC BY-SA 3.0 |
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| Jan 12, 2012 at 11:10 | comment | added | Emil Jeřábek | I suppose products of two fields do not count? | |
| Jan 12, 2012 at 10:53 | history | asked | Alan Haynes | CC BY-SA 3.0 |