More Abstract Algebra!

[headf1rst13]

I have not posted in a while, but I have some more questions

Comments

[joshua_green]

1. I think it's going to be difficult to make matrices work for this one, though they are standard things to keep in mind.  I'd consider one of the standard extensions of C.

2. What must be in R?  What does that generate?

3. What must be in any subdomain of Z?  What does that generate?

4. What must be in any subdomain of Z/(pZ)?  What does that generate?

[mathic]

I think most of the standard matrix groups work, no? Granted it may be difficult to show without knowledge of determinants.

[joshua_green]

What standard matrix groups are closed under addition?

[mathic]

Ha, I knew I was missing something.. however I can still think of two :)

Less standard.. but as long as you pick some well placed zeros, it's closed under addition and multiplication

[headf1rst13]

1. H, the ring of quaternions. Cool. Thanks.

2. Gotcha... thanks.

3. Had that from the previous comment. But thank you too.

4. Still confused... the a subdomain of Z/(pZ) must have the element 1, and this will generate all of Z/(pZ)... but I still don't know how to show that there are no other subdomains besides itself. I mean clearly a subset of Zp is not going to be closed under the operations mod p, so no subsets can be subdomains, but I'm not sure what you are getting at.

[joshua_green]

For 4, you're already done!  You've shown that any subdomain must contain 1, hence contain all of Z/(pZ), hence equal Z/(pZ).