In connection with [this problem](https://mathoverflow.net/questions/271121/an-optimization-problem-in-finite-groups): 



> 
> Do there exist integers $a\_0,\dotsc,b\_{10}\ge 0$ such that $a\_0+\dotsb+a\_{10}=36$, $b\_0+\dotsb+b\_{10}=37$, and 
>  $$ (a\_0+a\_1\zeta+\dotsb+a\_{10}\zeta^{10})(b\_0+b\_1\zeta+\dotsb+b\_{10}\zeta^{10})=1, $$ where $\zeta$ a primitive $11$-th root of unity? If they exist, can one explicitly describe / list all of them?
> 
> 
> 


Notice that $36\cdot 37=11^3+1$.



Special units in the $11$th cyclotomic field
Yes. Indeed
$$
(1 + \zeta + \zeta^{10}) \, (\zeta + \zeta^4 + \zeta^7 + \zeta^{10}) = 1
$$
with $\sum\_i a\_i = 3$ and $\sum\_i b\_i = 4$; now change each $a\_i$ to $a\_i+3$
and each $b\_i$ to $b\_i+3$. (This is not the only solution:
$(2+\zeta+\zeta^{-1})^{-1}$ also works with some room to spare.)


