Find the quotient field $\displaystyle K$ of $\displaystyle \mathbb{Z}[2i]$ and the integral closure of $\displaystyle \mathbb{Z}[2i]$ in $\displaystyle K$.

($\displaystyle i=\sqrt{-1}$)

Printable View

- Mar 20th 2011, 09:37 PMKaKaintegral closure of Z[2i]?
Find the quotient field $\displaystyle K$ of $\displaystyle \mathbb{Z}[2i]$ and the integral closure of $\displaystyle \mathbb{Z}[2i]$ in $\displaystyle K$.

($\displaystyle i=\sqrt{-1}$) - Mar 21st 2011, 02:28 AMTheArtofSymmetry
It is easy to see that K=Q(i). Now, is $\displaystyle i \notin \mathbb{Z}[2i]$ integral over $\displaystyle \mathbb{Z}[2i]$? Think about $\displaystyle x^2-2ix-1 \in \mathbb{Z}[2i][x]$. Can you see that the integral closure of $\displaystyle \mathbb{Z}[2i]$ in $\displaystyle K$ is simply $\displaystyle \mathbb{Z}[i]$? .