On Weil's proof of the bound for Kloosterman sums. (Q1867451)

For a finite field \(\mathbb{F}_q\) of characteristic \(p\) and \(a\in \mathbb{F}_q^*\) the corresponding Kloosterman sum \[ \text{Kl}(q,a)=\sum_{x\in \mathbb{F}_q^*} \text{e}^{2\pi i \text{\,Tr}(x+a/x)/p}, \] where Tr is the trace from \(\mathbb{F}_q\) to \(\mathbb{F}_p\), satisfies Weil's bound \(| \text{Kl}(q,a)| \leq 2\sqrt{q}\). While most proofs of this bound are valid for arbitrary \(p\), Weil's original proof assumes \(p\neq 2\) and uses Salié's formula \[ \text{Kl}(q,a)=\sum_{y\in \mathbb{F}_q} \eta(y^2-4a) \text{e}^{2\pi i \text{\,Tr}(y)/p}, \] where \(\eta\) is the quadratic character of \(\mathbb{F}_q\). The author extends Weil's method to arbitrary \(p\) by defining a quadratic character \(\Psi_c\) on the group \((\mathbb{F}_q[T]/T^2(T^2-c))^*\), \(c\in \mathbb{F}_q^*\). He also applies his approach to twisted Kloosterman sums \[ \text{Kl}(q,a,\chi)=\sum_{x\in \mathbb{F}_q^*} \chi(x) \text{e}^{2 \pi i \text{\,Tr}(x+a/x)}, \] where \(\chi\) is a nontrivial multiplicative character of \(\mathbb{F}_q\).