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The evaluation of Gauss sums for characters of 2-power order. (English) Zbl 1030.11067
Let \(p\) be a prime number, \(N \geq 2\) a positive integer not divisible by \(p\), \(k= \text{ord}_{N}(p)\) the order of \(p\) in \(( \mathbb{Z}/N \mathbb{Z})^{*}\), and \( \chi\) a multiplicative character of order \(N\) of the finite field \(F_{p^{k}}\). For \(1 \leq r \leq N-1\) and \(1 \leq v \leq p-1\) the Gauss sum \(g( \chi^{r},v)\) related to \( \chi^{r}\) and \(v\) is defined by \[ G( \chi^{r},v)= \sum_{x \in F_{p^{r}}} \chi(x) \zeta_{p}^{v Tr(x)}, \] where \( \zeta_{p}= \exp(2 \pi i/p)\) and \(Tr\) is the trace map from \(F_{p^{k}}\) onto \(F_{p}\). In a number of cases the Gauss sums \(G( \chi^{r},v)\) were determined up to complex conjugation. The authors consider the special case \(N=2^{t}\) not investigated earlier. Since \(G( \chi^{r},v)= \overline{ \chi^{r}(v)}G( \chi^{r},1)\) it suffices to consider the sum \(G( \chi^{r})=G( \chi^{r},1)\).
The arguments of the paper can be shortly described as follows. To evaluate \(G( \chi^{r})\) the authors study the tower of fields \[ \mathbb{Q} \subset \mathbb{Q}(i\sqrt{2}) \subset \mathbb{Q}( \zeta_{N}) \subset \mathbb{Q}( \zeta_{N}, \zeta_{p}), \] where \( \zeta_{N}= \exp(2 \pi i/N)\), and then apply Stickelberger’s theorem on prime decomposition of Gauss sums. Clearly, \(G( \chi^{r}) \in \mathbb{Q}( \zeta_{N}, \zeta_{p})\). In the cyclotomic field \( \mathbb{Q}( \zeta_{N})\) the principal ideal \((p)\) can be written as \( { \mathfrak {p}}_{1}{ \mathfrak{p}}_{2}\). Given a prime ideal \({ \mathfrak{p}}_{1}\) in the ring of integers \( \mathfrak{O}( \zeta_{N})\) of \( \mathbb{Q}( \zeta_{N})\) one can identify \( \mathfrak{O}( \zeta_{N})/{ \mathfrak{p}}_{1}\) with \(F_{p^{k}}\), where \(k= \text{ord}_{N}(p)= \phi(N)/2=2^{t-1}\). If \(p \equiv 3 \pmod 8\) and \(N=2^{t}\) with \(t \geq 3\), firstly the authors prove that \(G( \chi)/i \sqrt{p} \in \mathbb{Q}(i \sqrt{2})\). Then they show that up to sign \[ G( \chi)=i \sqrt{p} \cdot p^{2^{t-3}-1}(a+ib \sqrt{2}), \] where \(a,b \in \mathbb{Z}\) are determined from \((a+ib \sqrt{2})= { \mathfrak{p}}_{1} \cap \mathfrak{O}(i \sqrt{2})\). The ambiguity of sign can be resolved by Stickelberger’s congruence. Making use of similar arguments, the authors evaluate \(G( \chi^{2^{s}})\) separately for \(p \equiv 3\pmod 8\), \(0 \leq s \leq t-1\) and for \(p \equiv 5 \pmod 8\), \(0 \leq s \leq t-1\).

MSC:
11T24 Other character sums and Gauss sums
11L05 Gauss and Kloosterman sums; generalizations
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