Durham, UK

Stream: Geographic locality

[Eric Rodriguez]

Just realised I never posted in this stream! Kind of timely, as I just finished my third (and last) year exams! :shock:

[Eric Rodriguez]

I am (was?) a BSc Maths&CS student here :)

[Kevin Buzzard]

What are you doing next year and what are your mathematical interests?

[Eric Rodriguez]

I'm not sure yet! Hopefully some sort of further studies, but I haven't applied yet... My main interest is computational complexity, but I still carry around a childhood love for number theory :)

[Kevin Buzzard]

We still don't know the theorem which states the rank of the unit group of a number field.

[Kevin Buzzard]

or more generally the rank of the S-unit group of a global field. Am I right @Anne Baanen ?

[Kevin Buzzard]

It makes the sense to "invert a prime ideal" in the integers of a number field, because the prime ideal has finite order in the class group and so one can invert instead a generator of some prime ideal of the class group. Note that this is not true as far as I can see for general Dedekind domains.

[Kevin Buzzard]

The theorem is that if S is a finite set of nonzero elements of a Dedekind domain with finite class group then inverting S is the same as inverting all the prime factors of all the elements of S. In the number field and function field case the unit group is finitely-generated and there is an explicit formula for its rank in terms of the number of prime ideals inverted and the number of infinite places (which are the other places where you allow elements of norm greater than one).

[Anne Baanen]

Kevin Buzzard said:

or more generally the rank of the S-unit group of a global field. Am I right Anne Baanen ?

Correct, we have not yet done much with the unit group, although that is of course an important further direction.

[Eric Rodriguez]

This looks really interesting - what source would you guys recommend for the maths proof of this? The course I did just barely scratched the surface of class groups, sadly, so I do want to learn more

[Kevin Buzzard]

I learnt this stuff from Marcus' book but there must be other sources these days