| Pure cubic fields ordered by ascending radicands (2002/03/04) |
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Dan (02/03/04):
With enormous computational expense and using the Voronoi algorithm [1] , I have calculated the 827600 pure cubic fields L = Q(R^1/3) with radicands R <= 10^6, their regulators and ambiguous principal ideals. By suitable queries in the corresponding database I found the following statistical results: Population of the Dedekind Types: DT 1A (v_3(f) = 2): 254254 fields (30.72 %) DT 1B (v_3(f) = 1): 382231 fields (46.19 %) DT 2 (v_3(f) = 0): 191115 fields (23.09 %) where f denotes the conductor of the normal field N of L over its quadratic subfield K. Population of the Principal Factorization Types: PFT ALPHA (relatively ambiguous principal ideals): 163530 fields (19.76 %) PFT BETA (absolutely ambiguous principal ideals): 635434 fields (76.78 %) PFT GAMMA (some unit in the normal field has norm [-1+(-3)^1/2]/2): 28636 fields (3.46 %) Remarks: In comparison to my previous table of 1989 [2] (top leader up to now), which contained data for 82264 fields with R < 10^5, Dedekind Types remain remarkably stable (the same percentages), whereas PFT Beta still shows increasing tendency (77% instead of 75%) and PFT Alpha (20% instead of 21%) and PFT Gamma (3% instead of 4%) decrease. The interesting phenomena of exotic fields will be covered in a later communication. Ordering pure cubic fields Q((m*n^2)^1/3) with respect to ascending radicands R = m*n^2 yields other statistics of Dedekind Types (DT) and Principal Factorization Types (PFT) than ordering them by ascending conductors f = [3*]m*n. Bibliography:
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Dan (02/03/04):
This was indeed not a piece of cake.
It is the most extensive computation of pure cubics of all times. In fact, to fully realize the amount of work, which is necessary to get these results, it is illuminating to compare the previous tables of cubics that used VORONOI's algorithm: I. O. Angell, 1972, 3169 complex cubics with d > -2*10^4 I. O. Angell, 1975, 4794 (correct 4804) totally real cubics with d < 10^5 P. Barrucand / H. C. Williams / L. Baniuk, 1976, 8122 pure cubics with R < 10^4 H. C. Williams, 1982, 12220 pure cubics with R < 1.5*10^4 V. Ennola / R. Turunen, 1985, 26440 totally real cubics with d < 5*10^5 G. Dueck / H. C. Williams, 1985, 24537(+1) pure cubics with R < 3*10^4 D. C. Mayer, 1989, 4885 complex cubics with d > -3*10^4 D. C. Mayer, 1989, 82264 pure cubics with R < 10^5 G. W. Fung / H. C. Williams, 1990, 181748 (correct 182417) complex cubics with d > -10^6 D. C. Mayer, 1991, 10015 totally real cubics with d < 2*10^5 D. C. Mayer, 2002, 827600 pure cubics with R < 10^6 The number 827600 clearly beats all earlier field counts by far, it even surpasses the following table that was not computed with Voronoi's algorithm and does not contain regulators, units, and class numbers: P. Llorente / J. Quer i Bosor, 1988, 592923(-1) totally real cubics with d < 10^7 However, it must be remarked that Karim Belabas has developed an incredibly powerful and fast algorithm to count arbitrary cubic fields and to provide generating polynomials for them (without computing the regulator and class number): K. Belabas, 1997, 6715824025 totally real cubics with d < 10^11 K. Belabas, 1997, 20422230540 complex cubics with d > -10^11 See K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237 |
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Dan (02/03/04): Explanations:
Among all absolutely cubic non Galois extensions of the rationals Q, the pure cubic fields L are distinguished and uniquely characterized by the fact that their normal fields N are ramified cyclic cubic relative extensions of the special imaginary quadratic field K_0 = Q((-3)^1/2) with discriminant d_0 = -3 and class number h_0 = 1. Being ring class fields modulo 3-admissible conductors f over K_0, the Galois hulls N of pure cubic fields are nodes in the Abelian 3-network over K_0. Since K_0 is simultaneously the cyclotomic field of the 3rd roots of unity, zeta = [-1+(-3)^1/2]/2, zeta^2, i. e., the ray class field modulo 3 over Q, the normal fields N of pure cubic fields are Kummer extensions of K_0. The conductor f of such a normal field N is uniquely determined by the normalized radicand R = m*n^2 (with square free coprime integers m > n >= 1) of the underlying pure cubic fieldīL = Q(R^1/3), since Dedekind's Thorem states that (*) f = 3*m*n, if R incongruent 1,8(mod 9) (fields of Dedekind Type 1), (**) f = m*n, if R congruent 1,8(mod 9) (fields of Dedekind Type 2). Conversely, however, there can be several normalized radicands R_1,...,R_v associated with a given (essentially square free) conductor f, since, in general, f = [3*]m*n can be split into m and n in several ways. We call the number v = m(f) of normalized radicands R_1,...,R_v, associated with the conductor f, the multiplicity of f. Our main interest concerns the pure cubic fields sharing a common conductor f. Definition. By a v-family of pure cubic fields we understand a multiplet (L_1,...L_v) of fields with the same discriminant d (resp. conductor f) and with strictly increasing normalized radicands R_1 < ... < R_v. Hence the members of the v-family are given by L_i = Q(R_i^1/3) for i = 1,...,v. Such a family of pure cubic fields is called complete, if it contains all pure cubic fields sharing the given conductor f. Definition. We declare the Principal Factorization Type of a family (L_1,...L_v) of pure cubic fields as the multiplet (tau_1,...tau_v) (up to permutations), where tau_i in { alpha, beta, gamma } denotes the PFT of the field L_i for i = 1,...,v. I got the idea of a large scale investigation of PFTs of families of pure cubic fields in 1990 (after similar first discoveries for non-pure complex cubics in 1989). But then this enterprise was completely out of my reach. It took more than 10 years that I arrived at this target now. The results will be discussed in a later communication. |