## Applications of p-adic gamma function to congruences of binomial coefficients |

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2.6 and theorem 2a mod 4.8 and lemma additive character adic numbers algebraic number Algebraic Number Theory character mod character of Fq character sum Congruences of Binomial corollary 3.7 cubic non-residue mod cubic residue mod define the p-adic definition denote discussion epZp equations in proposition Euler's criterion famous Davenport-Hasse theorems finite field follows immediately Fq given Fq to Fp G xr Gauss sum Gross-Koblitz formula Legendre symbol lemma 4.13 give Let q m-l m-l Ming Yeung mod p2 modp modp"+1 multiplicative character natural number norm Number Theory odd prime p-adic gamma function p-adic integers positive integer power residue symbol prime ideal proposition 4.7 proves our assertion quadratic residue mod quartic residue mod rational integers residuacity root of unity sextic Jacobi sums study binomial coefficients sums and Jacobi Suppose Teichmuller character Teichmuller representative theorem 2.9 trace from Fq University of California xr,x zero mod