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Kloosterman sums for prime powers in P-adic fields

Stanley J. Gurak (2009)

Journal de Théorie des Nombres de Bordeaux

Let K be a field of degree n over Q p , the field of rational p -adic numbers, say with residue degree f , ramification index e and differential exponent d . Let O be the ring of integers of K and P its unique prime ideal. The trace and norm maps for K / Q p are denoted T r and N , respectively. Fix q = p r , a power of a prime p , and let η be a numerical character defined modulo q and of order o ( η ) . The character η extends to the ring of p -adic integers p in the natural way; namely η ( u ) = η ( u ˜ ) , where u ˜ denotes the residue class...

Kummer congruences for expressions involving generalized Bernoulli polynomials

Glenn J. Fox (2002)

Journal de théorie des nombres de Bordeaux

We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.

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