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Suppose that K = $mathbbQ(zeta )$ where $zeta $ is a primitive nth root of unity, then the matching totally real subfield is $textK^+$ = $mathbbQ(zeta +text 1/zeta )$ . (These two have ‘matching’ rings of integers given by $mathbbZ[zeta ]$ and $mathbbZ[zeta +1/zeta ]$ .)

So the ‘canonical’ generator of $textK^+$ is $zeta +1/zeta =lambda _n=2operatornameCos(2pi /n)$ which has minimal degree $varphi (n)/2$ for n > 2, but any Galois conjugate of $lambda _n$ would suffice so $mathbbQ(zeta +1/zeta )=mathbbQ(zeta ^k+1/zeta ^k)$ for gcd(k,n) = 1. The question is how to characterize the possible generators of $textK^+$ .

For example $tan ^2(pi /n)$ will always be a generator of $textK^+$ based on the ‘double-angle’ formula relating cos and tan. There are other possible candidates including any algebraic integer a in $mathbbZ[zeta + 1/zeta ]$. Note that $tan ^2(pi /n)$ is not an algebraic integer when n is twice-odd (a case that some authors omit). For example $tan ^2(pi /6)$ is 1/3 but here $textK^+$ is $mathbbQ$ which is also $mathbbQ(1/3)$ . It can be awkward doing calculations with non-integer generators, and there are often advantages to using unit generators. Even $lambda _n$ sometimes fails here. For example $lambda _12$ =$sqrt3$ which is the ‘canonical’ choice of generator but it is not a unit and $tan ^2(pi /12)$ = $7-4sqrt3$ is a unit which can be used as an alternate generator of $textK^+$ for n =12. (Geometrically this ‘scaling parameter’ has the advantage of properly describing how generations of polygons scale relative to the dodecagon as ‘parent’. For n = 6 above, this natural geometric scaling is the identity so in this case $lambda _6$ = 1 is the correct scaling and $tan ^2(pi /n)$ should only be used as an alternative generator when n is twice-even.) The real issue is understanding what choices are possible and any help along these lines will be appreciated.

For $p$ prime thus for $n$ square-free the conjugates of $zeta_n$ are a normal basis so it is simply

$$mathbbQ(zeta_n) = sum_m in (mathbbZ/nmathbbZ)^timeszeta_n^m mathbbQ,qquad mathbbQ(zeta_n+zeta_n^-1) = sum_m in (mathbbZ/nmathbbZ)^times/ pm 1(zeta_n^m +zeta_n^-m) mathbbQ$$ $$mathbbQ(zeta_n+zeta_n^-1) = mathbbQ(beta), beta =!! !!sum_m in (mathbbZ/nmathbbZ)^times/ pm 1!!!!(zeta_n^m +zeta_n^-m) b_m iff forall k in (mathbbZ/nmathbbZ)^times/ pm 1, exists m, b_mk ne b_m$$

Not sure if there is a way to enumerate the $beta$ such that $mathbbZ[zeta_n+zeta_n^-1]= mathbbZ[beta]$ other than enumerating the elements of $mathbbZ[zeta_n+zeta_n^-1]$ and checking one by one