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yoyobarn
 Post subject: Is Z/nZ a principal ideal domain?
PostPosted: Fri, 2 Mar 2012 03:08:21 UTC 
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Hi, is Z/nZ a principal ideal domain?

How would we prove it?

Thanks a lot for help.
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outermeasure
 Post subject: Re: Is Z/nZ a principal ideal domain?
PostPosted: Fri, 2 Mar 2012 05:36:13 UTC 
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yoyobarn wrote:
Hi, is Z/nZ a principal ideal domain?

How would we prove it?

Thanks a lot for help.


Correspondence of ideals gives Z/nZ is a principal ideal ring, and the only thing left is for you to decide if it is a domain.

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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