On Euclid's Algorithm in Some Cyclic Cubic Fields
نویسنده
چکیده
We now let # be the set of points P o such that M = M(P0); if sup M(Pt) < M Pit* (which happens in all cases known so far) we call this the second minimum, M2. R(0) is Euclidean if, and only if, M{P) < 1 for all rational triads [x,y,z]; and if 4> and \jj can be expressed as quadratic polynomials in 0, then the field is said to be cyclic. Heilbronn [1] has shown that Euclid's Algorithm holds in only a finite number of cyclic cubic fields, and using his method I have found that the only possible Euclidean cyclic cubic fields of discriminant A < 10 are those of discriminant 7, 9, 13, 19, 31, 37, 43, 61, 67, 73, 103, 109, 127, and 157. Davenport [2] has considered the first two and gives the values of M a s 1/7 and 1/3 respectively, and Godwin [3] has shown that the next two are Euclidean. Using the method of Barnes and Swinnerton-Dyer [4] as extended by Samet [5], I have found the values of M for the fields of discriminants 13, 19, 31, 37, 43 and 73 and shown that the fields of discriminants 61 and 67 are Euclidean. The values of M2 for the fields of discriminant 19, 31 and 43 are also given. The remaining four fields remain undecided.
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تاریخ انتشار 2006