In the early years of algebraic number theory, different mathematicians built the theory in terms of different objects, and according to different rules, some seeking always to demonstrate that the objects were computable in principle. Later, prominently in the era in which electronic computers were becoming available for academic research, efforts were initiated by some to compute the objects of the theory in practice. By examining writings, research, and correspondence of mathematicians spanning these early and late computational periods, we seek to demonstrate ways in which ideas from the old tradition influenced the new. Among the connections we seek are personal influence on problem selection, and borrowing of computational methods. In particular, we examine such links among the works of Kurt Hensel, Helmut Hasse, Olga Taussky, and Hans Zassenhaus.