Timeline for Does every finite abelian $p$-group $G$ admit a local ring structure with residue field of the same rank as $G$?
Current License: CC BY-SA 3.0
21 events
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| Mar 25, 2017 at 14:19 | comment | added | Taras Banakh | @tj_ Sorry, I had no intention to offend you. Very often, in mathematics, too, people have some illusions. So, I had an illusion that I know how to define a multiplication turning the abelian group into a local ring with high ranked residue field. And I was almost sure that it is true, so just wanted to find a reference in order to save time and not write a proof which (as I thought) should exist. When you asked about the construction I referred to Wiki having in mind that the argument should be similar to the standard one. So, this is a true story. | |
| Mar 25, 2017 at 14:03 | comment | added | tj_ | ... If you were true, you would have said, that you actually don't know such a construction for arbitrary abelian p-groups, instead of presenting that I asked a beginners question! | |
| Mar 25, 2017 at 14:02 | comment | added | tj_ | @Taras Banakh: Let me say that I find your behavoir very, very rude. In the original version of your question you said there seems to be a standard construction turning an arbitrary abelian p-group into a local ring. It was this general construction, I asked for in a comment. Afterwards you changed your text simply saying there is a standard construction for elementary abelian p-groups (which is absolutely trivial). Then, you responded to my comment and refered to a Wiki article for the construction of finite fields. I know how finite fields are constructed. ... | |
| Mar 25, 2017 at 13:35 | vote | accept | Taras Banakh | ||
| Mar 25, 2017 at 11:36 | answer | added | Jeremy Rickard | timeline score: 6 | |
| Mar 25, 2017 at 9:28 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Changed "quotient field" to "residue filed" in the title.
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| Mar 25, 2017 at 9:14 | history | edited | Taras Banakh | CC BY-SA 3.0 |
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| Mar 25, 2017 at 9:12 | comment | added | Taras Banakh | @Matematicos-Chibchas The rank $rank(F)$ of a finite field $F$ is defined as the rank of its additive group, i.e. $F$ has cardinality $p^{rank(F)}$ for some prime number $p$. It may happen that in the Field Theory this rank is called differently (for example, dimension)? | |
| Mar 25, 2017 at 9:05 | history | edited | Taras Banakh | CC BY-SA 3.0 |
Changed order of Remarks.
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| Mar 25, 2017 at 9:05 | comment | added | Matemáticos Chibchas | Sorry for my ignorance, but how is defined the rank of a field? | |
| Mar 25, 2017 at 8:56 | history | edited | Taras Banakh | CC BY-SA 3.0 |
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| Mar 25, 2017 at 7:42 | comment | added | Taras Banakh | @YCor I had in mind cyclic $p$-groups (and corrected this place). Thanks. | |
| Mar 25, 2017 at 7:41 | history | edited | Taras Banakh | CC BY-SA 3.0 |
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| Mar 25, 2017 at 7:38 | comment | added | YCor | It's not important here, but it's not true (as you say in your definition) that the decomposition of finite abelian groups as product of cyclic groups is unique, think of $Z/6Z$. Yet it's true for finite abelian $p$-groups. | |
| Mar 25, 2017 at 7:15 | comment | added | Taras Banakh | @tj_ The (stanard) construction of a Galois field of cardinality $p^k$ is given in Wikipedia en.wikipedia.org/wiki/… | |
| Mar 25, 2017 at 7:13 | comment | added | Taras Banakh | @YCor I added the definition of a rank. | |
| Mar 25, 2017 at 7:11 | history | edited | Taras Banakh | CC BY-SA 3.0 |
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| Mar 25, 2017 at 5:43 | comment | added | tj_ | Can you please describe the standard construction you mentioned in remark 1. Thanks. | |
| Mar 25, 2017 at 4:35 | comment | added | YCor | I guess what you mean by rank of $G$ is the dimension of $G/pG$ over the field on $p$ elements, which is also the minimal number of generators of $G$. | |
| Mar 25, 2017 at 4:33 | history | edited | YCor |
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| Mar 24, 2017 at 21:17 | history | asked | Taras Banakh | CC BY-SA 3.0 |