## Stream: Geographic locality

### Topic: Durham, UK

#### Eric Rodriguez (May 29 2021 at 11:52):

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

#### Eric Rodriguez (May 29 2021 at 11:52):

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

#### Kevin Buzzard (May 29 2021 at 16:32):

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

#### Eric Rodriguez (May 29 2021 at 17:43):

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 (May 29 2021 at 18:35):

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

#### Kevin Buzzard (May 29 2021 at 18:36):

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

#### Kevin Buzzard (May 29 2021 at 18:38):

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 (May 29 2021 at 18:41):

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 (May 29 2021 at 18:46):

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 (May 30 2021 at 12:29):

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 (May 30 2021 at 13:28):

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

Last updated: Aug 05 2021 at 04:14 UTC