On -unramified extensions over imaginary quadratic fields

Abstract

AbstractLet $n$ be an integer congruent to $0$ or $3$ modulo $4$ . Under the assumption of the ABC conjecture, we prove that, given any integer $m$ fulfilling only a certain coprimeness condition, there exist infinitely many imaginary quadratic fields having an everywhere unramified Galois extension of group $A_n \times C_m$ . The same result is obtained unconditionally in special cases.