Register to reply

The isomorphism Q<g>

by Somewheresafe
Tags: isomorphism, q<g>
Share this thread:
Sep26-11, 07:02 AM
P: 4
HI everyone! Sorry to be bothering you again with another question. >.<

Anyway I think it's pretty well-known that

[itex]\mathbb{Q}\left\langle q\right\rangle\simeq\oplus_{d|n}\mathbb{Q}\left( \zeta_d \right)[/itex]

where n is the order of g (say in a group) and d the divisors of n.

I was kinda wondering if the same goes for

[itex]\mathbb{Z}\left\langle q\right\rangle\simeq\oplus_{d|n}\mathbb{Z}\left[\zeta_d \right][/itex]

What I do know is that, for the first isomorphism, the isomorphism was shown by using a lot of isomorphisms (first is regarding group rings over cyclic groups, second by the CRT, and lastly by Kronecker's Theorem, something like that). Can the second statement not be established by using these three isomorphisms? (I think it may fail for the third; ie, Kronecker's, since [itex]\mathbb{Z}[/itex] is not a field, but I'm not quite sure if there's a version of that theorem for rings which are not fields).

Thanks! :D
Phys.Org News Partner Science news on
Physical constant is constant even in strong gravitational fields
Montreal VR headset team turns to crowdfunding for Totem
Researchers study vital 'on/off switches' that control when bacteria turn deadly

Register to reply

Related Discussions
Proof: Show that R is not isomorphic to R* Calculus & Beyond Homework 11
Z_2 /<u^4+u+1> isomorphism Z_2 /<u^4+u^3+u^2+u+1> Calculus & Beyond Homework 1
Complete the group as isomorphic Calculus & Beyond Homework 5
Please explain isomorphism with respect to vector spaces. Linear & Abstract Algebra 4
Isomorphism with GLn(R) Linear & Abstract Algebra 5