Timeline for Non-abelian divisible groups
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 21, 2013 at 23:09 | comment | added | Salvo Tringali | @Misha. I agree, but my point is that I don't know, among the many things that I don't know, who was the first to address the question explicitly (just to have a trustful reference for all practical purposes). However, I guess that I would be better to give up with this, for it seems hard, and perhaps even pointless, to track back the paternity of the result. Thank you, in any case, for sharing your thoughts. | |
| Mar 21, 2013 at 22:45 | comment | added | Misha | Salvo: You just need to know is that if $A\in U(n)$, then eigenvectors with distinct eigenvalues are orthogonal (this was probably known to Hermite or Jordan). It then follows that $A$ is unitarily conjugate to a diagonal matrix $Diag(e^{it_1},...,e^{it_n})$, which gives you all the roots you need. I assume, Ore and Niven were doing something more advanced. | |
| Mar 21, 2013 at 18:58 | comment | added | Salvo Tringali | OK, I've finally given a look at Niven's 1941 paper. It deals with a more general question, i.e. sort of a fundamental theorem of algebra for polynomials with coefficients in the skew-field of Hamilton's quaternions. And Niven reports a remark of Jacobson according to which this result is a consequence of previous work by Ore, dating back to 1933, on non-commutative polynomials. | |
| Mar 21, 2013 at 18:01 | comment | added | Salvo Tringali | From the paper mentioned in my answer: "The existence of an $m$-th root of a quaternion $a$ is known", and then a note refers the reader to: I. Niven, Equations in quaternions, AMM, Vol. 48, 1941, pp. 654-661. Maybe you're right, but this is the only reference that I've found so far, and it dates back to 1941. | |
| Mar 21, 2013 at 16:56 | comment | added | Misha | Divisibility of SO(3) was known by mid-19th century (since every element of SO(3) is a rotation). Divisibility of SU(2) is also quite trivial (use normal form for unitary matrices). I do not think Niven should be credited with this. | |
| Mar 21, 2013 at 12:34 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Fixed a mistake carried over from the other thread; deleted 75 characters in body
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| Mar 21, 2013 at 11:17 | comment | added | Salvo Tringali | To clarify: This doesn't count as an answer to the specific question in the OP, but it is an attempt to suggest a direction, and it was too long for a comment. | |
| Mar 21, 2013 at 10:47 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
Added a reference
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| Mar 21, 2013 at 10:20 | history | answered | Salvo Tringali | CC BY-SA 3.0 |