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, 


1. root $x = \sum\_{k=1}^{h} (\zeta\_{p})^{a^k}$
2. $4<h<p-1$
3. degree $\frac{p-1}{h}$
4. 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*](http://www.math.uni-duesseldorf.de/~klueners/minimum/). Are there others? (I've searched small $p$ and there are none.)



Minimal polynomial of sums of roots of unity with constant term $\pm1$
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 $a$:


1. $C$ is a subgroup of $(\mathbf{Z}/p\mathbf{Z})^\*$ with $4<\#C<p-1$
2. 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.


