Unramified non-solvable extensions of algebraic number fields have not been investigated well. One has no general method examining the existence of such fields over some fixed ground field. We deal with a special type of equations \(X^n-aX+b=0\), and obtain unramified non-solvable extensions of algebraic number fields, i.e., we have: ``Let \(A_n\) be the alternating group of degree \(n\) Then there exist infinitely many quadratic number fields which have unramified Galois extensions with Galois groups \(A_n\).'' This assertion also holds substituting \(S_n\) for \(A_n\). The above result has also been obtained independently by \textit{Y. Yamamoto} [Osaka J. Math. 7, 57--76 (1970; Zbl 0222.12003)] For Part I, see Tôhoku Math. J., II. Ser. 22, 138--141 (1970; Zbl 0209.35602).