The notion of a computable function or relation in the domain of natural numbers is by now standard, and the fact that it is explicated correctly by the notion of recursivity (Church's thesis) is no longer open to doubt. Even so it is an intriguing philosophical problem to what category exactly this notion belongs (e.g., depending on one's school of thought, analytic, synthetic a priori, theoretical, empirical) Let me begin this talk by drawing your attention to the fact that the notion of computability in algebra is less clear, and that here it is even not obvious whether we are aiming at the explication of an objectively given notion, or at the description of the various activities of a number of individuals which they considered to be "effective" or "realizable in a finite nymber of steps".
A major figure in the history of effective methods in algebra was Kronecker. Among other things, he proposed J7J a method by which the reducibility of a polynomial of one variable with rational coefficients can be tested, and another by which the reducibility of polynomials of several variables is reduced to the reducibility of polynomials in a single variable. In a more advanced area he showed how to determine, effectively, the irreducible components of an algebraic variety (see below). However, the formal tools available in Kronecker's time precluded a precise determination of the notion of effectiveness, even if he had been disposed philosophically to embark on such an enterprise.