Studies on the Jacobi sum and its application to the Leopoldt conjecture

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Number theory has been developed relating closely to many areas in mathematics, and recently it is applied to physics and engineering. In the present research, we researched from the integrated standpoint. Investigators in this project attended related conferences, discussed the problem with related researchers, collected many related references, and analyzed the problem using computers. In the process of our research we realized the importance of studies on Gauss sums and Jacobi sums, namely we firmly believed the relation between the Leopoldt conjecture and Gauss sums. The Leopoldt conjecture says that units, which are multiplicatively independent over the ring of rational integers, of a finite algebraic number field are also multiplicatively independent over the ring of p-adic integers. It is very important conjecture and is still a very difficult open problem.This conjecture is equivalent to the nonvanishing of L function at 1. On the other hand, head investigator in the present research gave an algebraic proof of nonvanishing of L function at 1 using Gauss sums under certain conditions. This implies close and deep relation between the Leopoldt conjecture and Jacobi sums. He also obtained partial affirmative answer to the conjecture from the different standpoint. It seems to be important to pursue the research considering the relation to cohomology theory, integral representation, structure of Galois groups of algrbraic number fields, ramification theory, theory of p-adic L functions, explicit formula for Hilbert norm residue symbols, and its non-abelization.