Abstract :
Let K/Q be a cyclic extension of degree n of prime conductor p. The field K over Q has a normal basis generated by the Gauss period. The prime p is totally ramified in the field K, p = πn. In this paper the congruence for Gauss period β ≡ k ∑ni=0(1/(ki)!)πi (mod πn + 1) is proved. At the end of the paper we give three examples. Let p ≡ 1 (mod 3), 4p = a2 + 27b2. In the first example the number a is determined. The second example is dealing with Wieferich congruence 2p − 1 ≡ 1 (mod p2). In the third example we count the number of the solutions of the congruence 1 + x + y ≡ 0 (mod p) where x, y are elements of the subgroup H, of the index 5, of the group (Z/pZ)*.
