A new result on the nonexistence of generalized bent functions is presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations. At the same time,a method is given which can be used to simplify the known results. Then we give the bounds and the meaning in algebraic number theory of the parameters in our results.
LIU FengMei 1,& YUE Qin 21 Department of Information Research,College of Information Engineering,Information Engineering University,Zhengzhou 450002,China
Two new results on the nonexistence of generalized bent functions are presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations, and examples satisfying our results are given.
For F = Q（√εpq）, ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2^n, the quartic residue symbol （1/q）4 and binary quadratic forms such as q^h（-2p）/^4 = x^2 ＋ 2py^2 where h（-2p） is the class number of Q（√-2p）. The results are very useful for numerical computations.
