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I was studying some properties of the set of $n$-th root of unity in $\mathbb{C}$. Let $\Omega_{n}$ be the set of primitive $n$-th root of unity.

For convenience, let $\zeta_p$ be a primitive $p$-th root of unity. Note that $\alpha = \zeta_5 + \zeta_5^4$ has degree 2 minimal polynomial. Also note that $\beta = \zeta_7^1+ \zeta_7^2 + \zeta_7^4$ has degree 2 minimal polynomial. Suddenly I have suspect if this is common phenomenon concerning the set of $p$th root of unity. I have searched the case $p = 11$ exhaustively to find a degree two algebraic number $\gamma = \zeta_{11} + \zeta_{11}^3 + \zeta_{11}^4 + \zeta_{11}^5 + \zeta_{11}^9$. But I have failed both to generalize and to find an exotic case. So my question is the following;

Let $p$ be a positive prime integer. Determine if there is a size $\frac{p-1}{2}$ subset of $\Omega_p$ so that the algebraic degree of the sum of the elements is $2$. (If not, determine if there is a size $\frac{p-1}{2}$ subset of $\Omega_p$ so that the algebraic degree of the sum of the elements is lower than $p$.)

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Answer : yes, and this is well-known. The set you're looking for is the set of all quadratic residues modulo $p$. Learn about Gauss sums

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