Gauss Sum of Index 4: (2) Non-cyclic Case.

Authors :: Jing Yang
Shi Xin Luo
Ke Qin Feng
Source :: Acta Mathematica Sinica; May2006, Vol. 22 Issue 3, p833-844, 12p, 2 Charts
Publication Year :: 2006
Abstract: Assume that m ⩾ 2, p is a prime number, (m,p(p - 1)) = 1, -1 ∉ (p) ⊂ (∤/m∤)* and [(∤/m∤)* : (p)] = 4. In this paper, we calculate the value of Gauss sum G(X) = Σ<subscript>x∈F*<subscript>q</subscript></subscript>X(x)ζ<superscript>T(x)</superscript><subscript>P</subscript> over ...<subscript>q</subscript>, where q = p<superscript>f</superscript>, f = φ(m)/4, X is a multiplicative character of ...q and T is the trace map from ...q to ...p. Under our assumptions, G(X) belongs to the decomposition field K of p in ℚ(ζ<subscript>m</subscript>) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non-cyclic case. [ABSTRACT FROM AUTHOR]