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Let p be a prime, m ≥ 2, and ( m, p( p – 1)) = 1. In this paper, we will calculate explicitly the Gauss sum $$ G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } } $$ in the case of [ (ℤ/ mℤ)* : 〈 p〉] = 4, and −1 ∉ 〈 p〉, where q = p f , $$ f = \frac{{\varphi {\left( m \right)}}} {4} $$ , χ is a multiplicative character of $${\Bbb F}$$ q with order m, and T is the trace map for $${\Bbb F}$$ q / $${\Bbb F}$$ p . Under the assumptions [ (ℤ/ mℤ)* : 〈 p〉] = 4 and −1 ∉ 〈 p〉, the decomposition field of p in the cyclotomic field ℚ(ζ m ) is an imaginary quartic (abelian) field. And G (χ) is an integer in K. We deal with the case where K is cyclic in this paper and leave the non–cyclic case to the next paper. [ABSTRACT FROM AUTHOR]

In this paper, we shall prove that for any orientable 3–manifold M, there is a link L = K ∪ K 1 ∪ K 2 ∪ K 3 with four components in M, such that the complement of L, say M L , contains separating essential closed surfaces of all positive genera. [ABSTRACT FROM AUTHOR]