Computing by Homomorphic Images

After explaining the general technique of computing by homomorphic images, the chinese remainder algorithm and the Hensel lifting construction are treated extensively. Chinese remaindering is first presented in an abstract setting. Then the specialization to Euclidean domains, in particular Z, K [y] and Z [y₁,..., y _r ] is treated. The lifting construction is first also presented in an abstract form from which Hensel’s Lemma derives by specialization. After introducing Zassenhaus’ quadratic lifting construction, again, the case of Z and Z, K [y] and Z [y1,…., yr] is considered. For both techniques, chinese remaindering as well as the lifting algorithms, a complete computational example is presented and the most frequent applications are discussed.