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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
##### Keywords:
finite fields; Gauss sums
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##### References:
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