Gauss Sums and Binomial Coefficients Original Research Article

Suppose p=tn+r is a prime and splits as image1image2 in image(image). Let q=pf where f is the order of r modulo t, χ=ω(q−1)/t where ω is the Teichmüller character on imageq, and g(χ) is the Gauss sum. For suitable τiset membership, variantGal(image(ζt, ζp)/image) (i=1, …, g), we show that ∏gi=1 τi(g(χ))=pα((a+b image)/2) such that 4ph=a2+tb2 for some integers a and b where h is the class number of image(image). We explicitly compute a mod(t/gcd(8, t)) and a mod p, in particular, a is congruent to a product of binomial coefficients modulo p.