Zulip Chat Archive

Stream: Is there code for X?

Topic: A finite division ring is a field

Kevin Buzzard (Jun 05 2020 at 13:26):

Do we have that a finite division ring is a field?

Chris Hughes (Jun 05 2020 at 13:27):

Yes

Chris Hughes (Jun 05 2020 at 13:27):

Sorry. We have that a finite integral domain is a field.

Kevin Buzzard (Jun 05 2020 at 13:31):

What I'm asking is harder. I guess I know some really stupid proof involving Brauer groups and Galois cohomology but there will be some more basic proof which doesn't need any of that machinery

Johan Commelin (Jun 05 2020 at 13:33):

The stupid proof exists. My non-constructive proof of that fact is that I remember seeing a talk about it in a first year student seminar.

Johan Commelin (Jun 05 2020 at 13:37):

I didn't read this text: https://www.imsc.res.in/~knr/past/salem1612.pdf but it is probably formalizable

Johan Commelin (Jun 05 2020 at 13:38):

Or this one: https://groupprops.subwiki.org/wiki/Every_finite_division_ring_is_a_field

David Wärn (Jun 05 2020 at 14:13):

Iirc this is one of the very first results in "Proofs from the Book", after Bertrand's postulate

Filippo A. E. Nuccio (Jun 09 2020 at 06:57):

Johan Commelin said:

Or this one: https://groupprops.subwiki.org/wiki/Every_finite_division_ring_is_a_field

This one is indeed quite basic, I studied it in my first year of undergraduate, taken from Herstein "Topics in Algebra": I knew nothing about cohomology (and almost nothing about Galois theory).

Johan Commelin (Jun 09 2020 at 07:01):

Do you want to try it?

Last updated: Dec 20 2023 at 11:08 UTC