Gauss sum of index 4. I: Cyclic case (Q2505358)

Let \(p\) be a prime number, \(m \geq 2\) an integer relatively prime to \(p(p-1)\), and \( \langle p \rangle\) the cyclic subgroup of \({( \mathbb Z}/{m \mathbb Z})^{*}\) generated by \(p\). Let \(\mathbb F_{q}\) be a finite field with \(q=p^{f}\) elements, \( \chi\) a multiplicative character of the field \(\mathbb F_{q}\) of order \(m\), and \(\text{tr}: \mathbb F_{q} \rightarrow \mathbb F_{p}\) the trace map. The authors calculate explicitly the Gauss sum \[ G( \chi)= \sum_{x \in F_{q}^{*}} \chi(x) e^{2 \pi i tr (x)/p} \] in the case when \([({ \mathbb Z}/{m \mathbb Z})^{*} : \langle p \rangle ]=4\), \(f= \frac{\varphi (m)}{4}\), \(-1 \in \langle p \rangle\), and the decomposition field of \(p\) in the cyclotomic field \({ \mathbb Q} ( \zeta_{m})\) is a cyclic quartic field.