Minimal polynomial of sums of roots of unity with constant term $\pm1$ Let prime $p$ and given $\zeta_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of  $x = \zeta_p + \zeta_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of four terms $x = \zeta_p +\zeta_p^{a}+\zeta_p^{p-a}+\zeta_p^{p-1}$ have as well. However, we also have,
$$F(x) = x^5 + x^4 - 28x^3 + 37x^2 + 25x + 1=0,\;\; x = \sum_{k=1}^{14} (\zeta_{71})^{23^k}$$
$$F(x) = x^6 + x^5 - 15x^4 - 28x^3 + 15x^2 + 38x - 1=0,\;\; x = \sum_{k=1}^{6} (\zeta_{37})^{11^k}$$
$$F(x) = x^7 + x^6 - 48x^5 + 37x^4 + 312x^3 - 12x^2 - 49x - 1=0,\;\; x = \sum_{k=1}^{16} (\zeta_{113})^{35^k}$$
$$F(x) = x^{11} + x^{10} - 40x^9 - 19x^8 + \dots - 1=0,\;\; x = \sum_{k=1}^{8} (\zeta_{89})^{12^k}$$
Question (edited):
What is the constraint on $p$ such that there is a minimal polynomial $F(x)$ with,   


*

*root $x = \sum_{k=1}^{h} (\zeta_{p})^{a^k}$ 

*$4<h<p-1$

*degree $\frac{p-1}{h}$

*and constant term $\pm1$? 


There are an infinite number of $p$ that satisfy conditions (1)-(3). But if we add the fourth, the four examples are the only ones in Kluener's Database of Number Fields. Are there others? (I've searched small $p$ and there are none.)
 A: This is not a complete answer, but a reformulation of your question in a way that removes the algebraic number theory, which translates the question into the realm of additive combinatorics.  I assume that you also require the order of $a$ in $(\mathbf{Z}/p\mathbf{Z})^*$ to be $h$, as in your examples and Noam's comment.  In this case, your conditions are equivalent to the following, where I write $C$ for the subgroup of $(\mathbf{Z}/p\mathbf{Z})^*$ generated by $a$:


*

* $C$ is a subgroup of $(\mathbf{Z}/p\mathbf{Z})^*$ with $4<\#C<p-1$

* writing $D_1,D_2,\dots,D_r$ for the distinct cosets of $C$ in $(\mathbf{Z}/p\mathbf{Z})^*$, the number of representations of $1$ as a sum $d_1+d_2+\dots+d_r$ with $d_i\in D_i$ differs by $1$ from the number of representations of $0$ as such a sum.


The reason for this is that the constant term of your minimal polynomial is (up to multiplication by $\pm 1$) the norm of your element $x$ from $\mathbf{Q}(\zeta_p)$ to $\mathbf{Q}$.  This norm is the product of the conjugates of $x$, and we can write down these conjugates.  Let $C$ be the subgroup of $(\mathbf{Z}/p\mathbf{Z})^*$ generated by your element $a$, and let $b_1,\dots,b_r$ (with $r:=(p-1)/h$) be representatives of the distinct cosets of $C$ in $(\mathbf{Z}/p\mathbf{Z})^*$.  Then the conjugates of $x$ are the elements
$$
\sum_{k=1}^h \zeta_p^{b_i a^k},
$$
and your question asks when the product of these $(p-1)/h$ numbers is $\pm 1$.  Equivalently,
$$
\pm 1 = \prod_{i=1}^r \sum_{c\in C} \zeta_p^{b_i c} = \sum_{c_1,\dots,c_r\in C} \zeta_p^{\sum_{i=1}^r b_i c_i}.
$$
This expression is a $\mathbf{Z}$-linear dependence on the $p$-th roots of unity, so it must have the form $n\cdot 1 + n\cdot\zeta_p+\dots+n\cdot\zeta_p^{p-1}=0$ for some integer $n$.  Thus, the collection of sums $\sum_{i=1}^r b_i c_i$ with $c_i\in C$ must consist of $n$ copies of each nonzero element of $\mathbf{Z}/p\mathbf{Z}$, together with either $n-1$ or $n+1$ copies of zero.  Finally, any two nonzero elements $u,v$ of $\mathbf{Z}/p\mathbf{Z}$ always have equal numbers of representations as $\sum_{i=1}^r b_i c_i$ with $c_i\in C$, since we get a bijection between the two sets of representations by multiplying all representations of $u$ by $v/u$.  Thus your condition is the same as asserting that the numbers of representations of $1$ and $0$ differ by $1$, which is the claimed reformulation.
In view of this reformulation, you might want to add a tag to your question in order to alert the additive combinatorialists, since it seems related to sum-product phenomena.
