A note of generalized bent functions

A relationship between the nonexistence of generalized bent functions and class groups

“…There exists s 1 1, such that p s1 1 ≡ −1 (mod p 2 p 3 ). Then if there exists ζ ∈ O K such that ζζ = (2N ) n , then there exists α ∈ O K such that αᾱ = (2p 2 p 3 ) n .…”

“…A function f : Z n q → Z q is called a generalized bent function (GBF) [1] if the equality x∈Z n q ζ f (x)−x·y q = q n/2 holds for every y ∈ Z n q . We call [n, q] the type of such GBF f .…”

“…Then o([Q 2 ]) = 2r 1 and o([P 2 ]) = 2s 2 in C(F 2 ), where r 2 and s 2 are odd integers.where ε = 0 or 1.Proof (1). In the class group C(F 1 ) of the field F1 = Q( √ −p 2 ), if [Q 1 ] a = [P 1 ] b where 2 a, then [Q 1 ] a [P 1 ] b = 1.…”

“…In the class group C(F 1 ) of the field F1 = Q( √ −p 2 ), if [Q 1 ] a = [P 1 ] b where 2 a, then [Q 1 ] a [P 1 ] b = 1. Without loss of generality, we let a > 0, b 0.…”

“…(1) Since that p 1 and p 3 are the only primes ramifying in F 2 , r 2 (C(F 2 )) = 1 and o([P 3 ]) = 2 by genus theory. For ( p1 p3 ) = −1, the equation p 3 z 2 = x 2 + p 1 p 3 y 2 has only trivial Z-solution by Lemma 1.4, i.e., p 3 / ∈ N F2/Q (F * 2 ).…”