On Gauss Sums (mod P～l) L≥2

In this thesis, we discuss the Gauss sumswhere p denotes odd prime, (n,p) = 1.χis a Dirichlet multiplicative charactermod pl. As to these sums, the Soviet Union mathematician I.M.Vinogradov gave thefollowing result in his book'basic number theory'(reference to[1]):In this thesis, we discuss the sums in detail and find that if G(n,χ) = 0, then|G(n,χ)| = 2 pl. This Gauss sums have attracted many people's attention(referenceto [2, 3]), but has not been solved completely. In the thesis, we obtain the followingresults: ifχif non-primitive, then G(n,χ) = 0 ; ifχis primitive, and for any characterλ,χ=λ2, then G(n,χ) = 0; ifχis primitive, and there exits a characterλ, such thatχ=λ2, then |G(n,χ)|2 =2pl(1 +ρ(nA)).Whereρis the quadratic character mod pl, A is an integer determined by the equationFurthormore, we obtain the value of G(1,χ) when .Where, . And we could use the Galois theory and theabove result to get the value of G(n,χ) for any primitive characters .