Jacobsthal sums of prime order (Q1084131)

Let \(p\) be an odd prime and let \(q=p^{\alpha}\), where \(\alpha\geq 1\). For \(a\in \mathrm{GF}(q)\) and \(e\geq 1\) the Jacobsthal sum \(\Phi_e(a)\) of order \(e\) is defined by \[ \Phi_e(a)=\sum_{x\in \mathrm{GF}(q)}\left(\frac{x}{q}\right)\left(\frac{x^e+a}{q}\right), \] where \((\frac{.}{q})\) is the Legendre symbol for \(\mathrm{GF}(q)\). The authors evaluate \(\Phi_e(a)\), for any odd prime \(e\), in terms of the solutions of an appropriate quadratic partition of \(q\).