Evans, Ronald J. Identities for products of Gauss sums over finite fields. (English) Zbl 0491.12020 Enseign. Math., II. Sér. 27, 197-209 (1981). 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. Reviewer: Saburô Uchiyama (Tsukuba) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 6 ReviewsCited in 19 Documents MSC: 11T24 Other character sums and Gauss sums Keywords:products of Gauss sums; finite fields PDFBibTeX XMLCite \textit{R. J. Evans}, Enseign. Math. (2) 27, 197--209 (1981; Zbl 0491.12020)