Timeline for Solving $z^n=a+bi$ using only radicals of positive real numbers
Current License: CC BY-SA 3.0
14 events
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| Dec 19, 2014 at 21:49 | comment | added | Alexander Kuleshov | Dear Hugo, right now I can't give such an answer, and I feel that this problem is very complicated even in case $n=3$. I suggest to concentrate on $3x-4x^3 =a$ equation. If we find a criterion for its "positive solvability" in terms of rationals and "$a$", we'll be there. At least for $a=\sin(y)$ where $y=3\pi/2,0,\pi/2, \pi/4, \pi/5, \pi/10, \pi/16, \pi/17, \pi/20, \pi/32,\pi/34, \pi/40, \pi/64...$ this is true. And what if $a=\sin(1)$? I don't know if there is a general solution to this problem. | |
| Dec 19, 2014 at 20:57 | comment | added | Hugo Chapdelaine | Dear Alexander, I think that what was on my mind when I wrote this post was given $n\in\mathbf{Z}_{\geq 3}$ and $a,b$ algebraic real numbers, can you give a criterion which says when is $z^{n}-(a+bi)$ positive solvable. | |
| Dec 19, 2014 at 20:54 | comment | added | Alexander Kuleshov | And the question for specific $a,b$ is much more complicated. Lets consider $n=3$ and formulate it in terms of $\varphi$. Then it will turn into: find all $a\in[-1,1]$ such that the roots of $3x-4x^3-a$ have algebraic (involving rational powers) expressions over rationals and "$a$". Here $a$ states for $\sin(\varphi)$, $x$ for $\sin(\varphi/3)$. We know that for $a=-1,0,1$ the answer is positive. What about all the others? | |
| Dec 19, 2014 at 19:25 | history | edited | Alexander Kuleshov | CC BY-SA 3.0 |
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| Dec 19, 2014 at 19:10 | comment | added | Alexander Kuleshov | Ooh I see! I probably misread your question: did you mean a criterion for every specific a,b,n? It's a different problem... I just found all $n$ such that $z^n=a+ib$ is positive solvable for all $a,b\in R$. | |
| Dec 19, 2014 at 18:53 | comment | added | Alexander Kuleshov | Dear Hugo, in my answer I mentioned that this article was already quoted "above". By "above" I meant exactly your post. And now I gave a complete answer to your question. Although this observation is based on the result from "radical tower and roots of unity", it required some additional analysis, now I think that my phrase "follows directly" is a bit incorrect. | |
| Dec 19, 2014 at 17:12 | comment | added | Hugo Chapdelaine | Dear Alexander, already in the answer that I had posted on Sep 21, 2012, I was quoting Brian Conrad's note: "radical tower and roots of unity" | |
| Dec 19, 2014 at 11:49 | history | edited | Alexander Kuleshov | CC BY-SA 3.0 |
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| Dec 19, 2014 at 11:35 | comment | added | Alexander Kuleshov | Done. Note that there are some inaccuracies in Weisstein's overview, but the key result is the same: $\sin\displaystyle\Big(\frac{\pi}{n}\Big)$ can be expressed in radicals(i.e. composition of algebraic operations and rational powers at rational numbers) iff $\varphi(n)=2^m$. | |
| Dec 19, 2014 at 11:25 | history | edited | Alexander Kuleshov | CC BY-SA 3.0 |
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| Dec 19, 2014 at 5:47 | comment | added | Gerry Myerson | "The answer to your question follows directly from the paper mentioned above." above is not constant across platforms and viewing modes. Could you please employ some more invariant term, so I can know what paper you mean? | |
| Dec 19, 2014 at 5:17 | history | edited | Alexander Kuleshov | CC BY-SA 3.0 |
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| Dec 18, 2014 at 21:05 | history | edited | Alexander Kuleshov | CC BY-SA 3.0 |
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| Dec 18, 2014 at 19:04 | history | answered | Alexander Kuleshov | CC BY-SA 3.0 |