**I. Comparison**


It doesn't seem to be well-known that the generic cubic (prominent in this [MO post](https://mathoverflow.net/questions/438594/)) for $C\_3 = A\_3$,


$$x^3-nx^2+(n-3)x+1 = 0$$


has the nice property that its roots $a,b,c$, if in correct order, obey,


$$(a^2b)^{1/3}+(b^2c)^{1/3}+(c^2a)^{1/3} = 0$$


(I only noticed this after I asked an [MO question](https://mathoverflow.net/questions/438035/) about the similar-looking Klein quartic $a^3b+b^3c+c^3a=0.$)


Since the generic cubic is intimately connected to the roots of unity for prime $p\equiv 1\,\text{mod}\; 6$, (the case $n=1$ yields $1^{1/7}$), naturally I got curious about its big sister the *Emma Lehmer quintic* which is for $p\equiv 1\,\text{mod}\; 10$, namely,


$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + 10)x +1=0$$


It turns out its roots $x\_k$ may have a similar property.




---


**II. Question:**


Analogous to the generic cubic, is it true that the Emma Lehmer quintic obeys, $$(a^4b^3c^2d)^{1/5} + (b^4c^3d^2e)^{1/5} + (c^4d^3e^2a)^{1/5} + (d^4e^3a^2b)^{1/5} + (e^4a^3b^2c)^{1/5} = 0$$ for the **correct** ordering of its roots $a,b,c,d,e$?


***Update 1:*** Thanks to Peter Taylor in the comments, and using the fact that $abcde = -1$, we can get rid of the fifth roots and get the equivalent but more elegant form,


$$\frac1{a}-\frac1{ab}+\frac1{abc}-\frac1{abcd}+\frac1{abcde} = 0$$


Or more generally (for $\mu$ an integer),


$$\frac{\mu}{a}-\frac{\mu^{2}}{ab}+\frac{\mu^{3}}{abc}-\frac{\mu^{4}}{abcd}+\frac{\mu^{5}}{abcde} = 0$$


where $\mu^5$ is the constant term of the quintic and the Lehmer quintic the special case $\mu=1$.


***Update 2:*** I've already tested the [*Hashimoto septic*](https://mathoverflow.net/a/255424/12905) which fortunately has a seventh power $\mu^7$ as its constant term and the analogous relation,


$$\frac{\mu}{a}-\frac{\mu^{2}}{ab}+\frac{\mu^{3}}{abc}-\frac{\mu^{4}}{abcd}+\frac{\mu^{5}}{abcde}-\frac{\mu^{6}}{abcdef}+\frac{\mu^{7}}{abcdefg} = 0$$


and among the $(p-1)! = 720$ permutation of its roots, at least 9 works.




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**III. Example**


Let $n=-1$. Then we get the quintic for $p=11$ and its roots,


$$x^5 + x^4 - 4x^3 - 3x^2 + 3x + 1 = 0$$


$$a,b,c,d,e = 2\cos\frac{2\pi k}{11}$$


with $k = 1, 4, 5, 2, 3$ as the correct order. One can then verify it obeys the relation in the question.


**P.S.** For $p=5$, there are $(p-1)! = 24$ permutations of its roots. It is easy for a computer to find the correct order for any $n$ that I tested. But does it hold true for ALL $n$?



A similar relationship between the generic cubic and the Lehmer quintic?
The map $s(r) = \frac{n+2 + nr - r^2}{1 + (n+2)r}$ cyclically permutes the roots. This map is given in [2], and I found it through the reference in [1]. It turns out to give the correct order.


Explicitly, if we pick one root $x\_1$ we have


\begin{eqnarray\*}
D &=& n^3 + 5n^2 + 10n + 7 \\
Dx\_2 &=& (n^2 + 4n + 4)x\_1^4 + (n^4 + 4n^3 + 4n^2 - n - 2)x\_1^3 + \\&& (-2n^5 - 14n^4 - 43n^3 - 76n^2 - 80n - 39)x\_1^2 + \\&& (n^6 + 9n^5 + 37n^4 + 89n^3 + 131n^2 + 107n + 36)x\_1 + \\&& (n^4 + 8n^3 + 26n^2 + 39n + 22) \\
%
Dx\_3 &=& (-2n - 3)x\_1^4 + (-2n^3 - 4n^2 - 3n - 2)x\_1^3 + \\&& (3n^4 + 14n^3 + 31n^2 + 41n + 24)x\_1^2 + \\&& (-n^5 - 7n^4 - 21n^3 - 36n^2 - 29n - 6)x\_1 + \\&& (-2n^2 - 7n - 6) \\
%
Dx\_4 &=& (-n^2 - 3n - 2)x\_1^4 + (-n^4 - 3n^3 - 2n^2 + n + 1)x\_1^3 + \\&& (2n^5 + 12n^4 + 33n^3 + 54n^2 + 53n + 23)x\_1^2 + \\&& (-n^6 - 8n^5 - 29n^4 - 62n^3 - 81n^2 - 59n - 18)x\_1 + \\&& (-n^5 - 7n^4 - 24n^3 - 47n^2 - 52n - 25) \\
%
Dx\_5 &=& (n + 1)x\_1^4 + (n^3 + 2n^2 + 3n + 3)x\_1^3 + \\&& (-n^4 - 4n^3 - 9n^2 - 14n - 8)x\_1^2 + \\&& (-n^4 - 7n^3 - 19n^2 - 29n - 19)x\_1 + \\&& (n^4 + 6n^3 + 16n^2 + 20n + 9)
\end{eqnarray\*}


and it is pure computation to show that $$x\_1 x\_2 x\_3 x\_4 x\_5 - x\_2 x\_3 x\_4 x\_5 + x\_3 x\_4 x\_5 - x\_4 x\_5 + x\_5 = 0$$


[Online computation with sagecell.sagemath.org](https://sagecell.sagemath.org/?z=eJyFk01v2zAMhs8LkP9AeIdZTpxWsrXD0OzYc-sNuxQ1YNRyI8yRPFkd1A377yPleEmGoj0oovi8_JDoVJsr8xm2cGP7Z2P3uukrbR7T21u2XCwXrepgSAP7tFy8c8o_OQOhlrACU4ss1CXkkIrM1AW6PuIucOeXmYkbQwWB1KBwBfKg43wWyiiUpBOTjnj5Qh7a537G835SEpHeoCyHkAUGF5By8sZA9Me7VGJzFfx-eOG216558Nqaa636Nq0Y6qsC1Zy0GPfjyXqtjE_xLTADi_mCQIjNcDyFYrIF2eVkF2TLyS7Rdtb6EY93ga8hCFwFrhKXvKd87-Fb0-u28Qr8rvH4o54_OAVN30OrR6_Ng18uBqexj16ZdFQ-jTkZW0MyKyC6EvZWxiib090l9nsCepo1bLdwCaofFSRdo_sEOusggDZT0P0hN94W5xoKHFoocWJBsvo3v5B_6PkDJyaIFcTKmW2JIUE_ejEI1SiWR57z6KISX-wrmVavdECsIFYSk8T4KSuJSWKcmDhlkhgnJogVB0b9TN3XOV2gzgnWeaydx3T5v_r_3zE_O63OwOySjEo0poURB6nAWKPAdjS16eFBj_BLObuOrk53fhcBDI3zBDEiJ8FmnmvydaesU3sYnP2pTBxx4FnAP0aRhTI7dHY8UWdHMLvk6SdxgyU7oA9DtQn7C_6_KP4=&lang=sage&interacts=eJyLjgUAARUAuQ==)


However, this isn't a very satisfying proof because it doesn't address the question of necessary and sufficient properties of the original polynomial. Does a suitable permutation of roots exist for all irreducible polynomials of odd (or maybe odd prime) degree whose Galois group is cyclic? The equivalence between the forms


$$
(x\_1^4 x\_2^3 x\_3^2 x\_4)^{1/5} + (x\_2^4 x\_3^3 x\_4^2 x\_5)^{1/5} + (x\_3^4 x\_4^3 x\_5^2 x\_1)^{1/5} + (x\_4^4 x\_5^3 x\_1^2 x\_2)^{1/5} + (x\_5^4 x\_1^3 x\_2^2 x\_3)^{1/5} = 0 \\
\frac1{x\_1} - \frac1{x\_1 x\_2}+\frac1{x\_1 x\_2 x\_3} - \frac1{x\_1 x\_2 x\_3 x\_4} + \frac1{x\_1 x\_2 x\_3 x\_4 x\_5} = 0 \\
x\_1 x\_2 x\_3 x\_4 x\_5 - x\_2 x\_3 x\_4 x\_5 + x\_3 x\_4 x\_5 - x\_4 x\_5 + x\_5 = 0
$$
and a couple of other variants relies on the constant coefficient of Emma Lehmer's quintic being $1$. If there is a generalisation, which variant does it require? Or does generalisation require a cyclic Galois group and a unit constant coefficient?




---


[1] Henri Darmon, [*Note on a polynomial of Emma Lehmer*](https://www.math.mcgill.ca/darmon/pub/Articles/Research/03.Lehmer/paper.pdf), Math. Comp. vol. 56, no. 194, April 1991, pp. 795-800.  

[2] Rene Schoof and Lawrence C. Washington, [*Quintic polynomials and
real cyclotomic fields with large class numbers*](https://www.ams.org/journals/mcom/1988-50-182/S0025-5718-1988-0929552-2/S0025-5718-1988-0929552-2.pdf), Math. Comp. vol. 50,
no. 182, April 1988, pp. 543-556.


