The Euclidean algorithm in algebraic number fields. (English) Zbl 0843.11046

This survey deals with the Euclidean algorithm and Euclidean minima in algebraic fields. Most work defines the Euclidean algorithm with respect to the norm function, but here other functions are also considered, as well as the \(k\)-stage algorithm first introduced by G. Cooke [J. Reine Angew. Math. 282, 133-156 (1976; Zbl 0328.13013) and ibid. 283/284, 71-85 (1977; Zbl 0343.13008)]. Particular attention is paid to pointing out where unsolved problems exist and these are also shown by the gaps in the extensive tables of results.
The bibliography contains 219 references, but unfortunately does not include all the papers cited in the survey, nor is it in strict alphabetical order, nor do the dates of papers always agree with the dates given in the survey. This is a pity, because it detracts a little from what is otherwise a valuable piece of work.
Reviewer: H.J.Godwin (Egham)


11R04 Algebraic numbers; rings of algebraic integers
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11H50 Minima of forms
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors