Counting Monogenic Cubic Orders

Monogenity and Power Integral Bases: Recent Developments

“…• M. Kang and D. Kim [132] considered the number of monogenic orders in pure cubic fields • J. H. Evertse [31] considered "rationally monogenic" orders of number fields • S. Akhtari [2] showed that a positive proportion of cubic number fields, when ordered by their discriminant, are not monogenic • L. Alpöge, M. Bhargava, A. Shnidman [4] showed that if isomorphism classes of cubic fields are ordered by absolute discriminant, then a positive proportion are not monogenic and yet have no local obstruction to being monogenic (that is, the index form equations represent +1 or −1 mod p for all primes p) • M. Bhargava [20] proves that an order O in a quartic number field can have at most 2760 inequivalent generators of power integral bases (and at most 182 if |D(O)| is suffciently large).…”

Monogenity and Power Integral Bases: Recent Developments

“…• M. Kang and D. Kim [122] considered the number of monogenic orders in pure cubic fields; • J. H. Evertse [123] considered "rationally monogenic" orders of number fields; • S. Akhtari [124] showed that a positive proportion of cubic number fields, when ordered by their discriminant, are not monogenic; • L. Alpöge, M. Bhargava, A. Shnidman [125] showed that, if isomorphism classes of cubic fields are ordered by absolute discriminant, then a positive proportion are not monogenic and yet have no local obstruction to being monogenic (that is, the index form equations represent +1 or −1 mod p for all primes p); • M. Bhargava [126] proved that an order O in a quartic number field can have at most 2760 inequivalent generators of power integral bases (and at most 182 if |D(O)| is sufficiently large). The problem is reduced to counting the number solutions of cubic and quartic Thue equations, somewhat analogously like described in Section 2.4, using a refined enumeration; • S. Akhtari [127] gave another proof of Bhargava's result [126]: she used the more direct approach of Section 2.4 and applied sharp bounds for the numbers of solutions of cubic and quartic Thue equations.…”

Quartic Index Form Equations and Monogenizations of Quartic Orders

“…The left-hand side of ( 17) is a cubic binary form in u and v whose coefficient are symmetric polynomials of ξ (1) , ξ (2) , ξ (3) , ξ (4) . Simple and routine calculations show that this integral cubic binary form is…”

“…where m ∈ Z. In [1] we have discussed some results about cubic Thue equations and their consequences in resolving index form equations and counting the number of monogenizations of a cubic ring.…”