That is a good question. Sage's number fields get their units and regulator by calling the corresponding functions in the pari library. As far as I can see the pari library does not have a function to compute th regulator of an arbitrary set of units. It would not be hard to implement this in Sage.
The ingredients you need are: * K.complex_embeddings() gives all the embeddings of K into CC (the complex numbers). You would need to eliminate one of ecah conjugate pair of embeddings. TODO: implement a flag to complex_embeddings() which only gives one of each pair. * Now just evaluate the embeddings on your units, take logs, construct the appropriate matrix of those and find its determinant. * To find the index of your units, divide their regulator by the field's regulator. I have skated over some details, like what to do if the number of your units is different from the unit rank. Harder TODO: given any unit and a Z-basis for the units, express your unit as a Z-linear combination of the generators. If that was implemented, then the answer to your original question would be a simple matter of finding the determinant of an integer matrix. These are all things which it would be good to have implemented in Sage. Feel free to do so and submit a patch! John Cremona 2008/9/3 Jannick Asmus <[EMAIL PROTECTED]>: > > Dear All, > > suppose that K is a number field and U the group of units in the maximal > order of K. Then the rank r of U, i.e. the rank r of the free group U_f= > U/Tor(U) (where Tor(U) denotes the group of torsion elements in U) is > given by Dirichlet's unit theorem. Clearly r is the dimension of the > Q-vector space U_Q = U (x)_Z Q. > > Sage gives a basis of U_Q (sage: K.units()). > > Now given r units u_1,...,u_r how can it be tested that the u's generate > U_Q - or are linearly independent over Q? > > If the structure of U_Q was additive, this might not be a problem for > SAGE as it is a standard problem in linear algebra boiling down to > calculate a determinant. But how to tackle this problem when the > structure of the Q-vector space is multiplicative, at least in notation. > > Alternatively we could consider the quotient group U/(u_1,...,u_r) and > test if it is of finite order. > > Thanks for any help. > > Please do not hesitate to ask for more information if something is > unclear or needs more information. > > Best wishes, > J. > > > ps: This is my first posting here. Hope that this email will get through > to the forum. > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
