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Identities for products of Gauss sums over finite fields. (English) Zbl 0491.12020

Several interesting identities are found and proved for Gauss sums defined over finite fields. With \(\zeta=\exp(2\pi i/p)\), \(p\) a prime, define the Gauss some over \(\mathrm{GF}(p^r)\) \((r\geq 1)\) by
\[ G(\chi)=G_r(\chi)=-\sum_{x\in\mathrm{GF}(p^r)} \chi(x)\zeta^{\mathrm{Tr}(x)} \] where \(\mathrm{Tr}\) is the trace map from \(\mathrm{GF}(p^r)\) to \(\mathrm{GF}(p)\), and \(\chi\) is any character on the multiplicative group of \(\mathrm{GF}(p^r)\) (with \(\chi(0)=0\)). Fix a prime power \(q=p^f\), \(f\geq 1\). Then we have
\[ 1=\frac{\chi^{\ell}(\ell)G_f(\chi)}{G_f(\chi^{\ell})}\prod_{j=1}^e \prod_{k=1}^r \prod_{c=1}^{w^{r-k}} \frac{G_{fn}\left(\chi^{(q^n-1)/(q-1)}\psi^{b(k,c,j)}\right)}{G_{fn}\left(\psi^{b(k,c,j)}\right)}\, , \] where \(\ell=w^r\) for a prime \(w\neq p\), \(n= \) the order of \(q\pmod w\), \(e=(w-1)/n\), \(b(k,c,j)=w^{k-1}(cw+i_j)\) with the coset representatives \(i_1,\dots,i_n\) for the cyclic subgroup \(\langle q\rangle\) in the multiplicative group of \(\mathrm{GF}(w)\), \(\chi\) is a character on \(\mathrm{GF}(q^n)\), \(\psi\) is a character of order \(\ell\) on \(\mathrm{GF}(q^n)\);
\[ 1=\frac{G_f(\chi^\alpha)}{\chi^\alpha(\ell) G_{f\ell}(\chi^{\alpha\beta})} \prod_{j=1}^{\ell-1} G_f\left(\chi^{j(q-1)/\ell}\right), \]
where \(\ell\mid q-1\), \((\alpha,\ell)=1\), \(\beta=(1+q+\dots +q^{\ell-1})/\ell\), and \(\chi\) is the character on \(\mathrm{GF}(q^{\ell})\) of order \(q^{\ell}-1\);
\[ \sum_{x,y\in\mathrm{GF}(q)} \chi_1(x,y)\chi_2((1-x)(1-y))\chi_3^2(x-y)=R(\chi_1, \chi_2, \chi_3)+ R(\chi_1, \chi_2, \chi_3\varphi), \] where \(\chi_1, \chi_2, \chi_3, \varphi\) are characters of \(\mathrm{GF}(q)\), \(\varphi\) has order 2 \((p>2)\), \(\chi_1 \chi_2 \chi_3^2\) and \((\chi_1 \chi_2 \chi_3)^2\) are nontrivial, and
\[ R(\chi_1, \chi_2, \chi_3)=\frac{G(\chi_3^2)G(\chi_1)G(\chi_1\chi_3)G(\chi_2)G(\chi_2\chi_3)} {G(\chi_3)G(\chi_1\chi_2\chi_3)G(\chi_1\chi_2\chi_3^2)}\, ; \] and
\[ \sum_{\substack{x,y\in\mathrm{GF}(q) \\ x,y\neq 0}} \chi_1\chi_3\left(\frac{1+x}{y}\right) \chi_2\chi_3\left(\frac{1+y}{x}\right) \chi_1\chi_2 (y-x)=D(\chi_1, \chi_2, \chi_3)+D(\chi_1\varphi, \chi_2\varphi, \chi_3\varphi), \] where \(\chi_1, \chi_2, \chi_3, \varphi\) are characters of \(\mathrm{GF}(q)\), \(\varphi\) has order 2 \((p>2)\), \(\chi_1^2,\chi_2^2, \chi_3^2, \chi_1\chi_2, \chi_1\chi_3\) and \(\chi_2\chi_3\) are nontrivial, and
\[ D(\chi_1, \chi_2, \chi_3)=\frac{q^2\chi_2(-1)G(\chi_1\chi_2\chi_3)}{G(\chi_1)G(\chi_2)G(\chi_3)}\, . \]
There are also some conjectural formulas proposed.

MSC:

11T24 Other character sums and Gauss sums
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