I'm having difficulties with a homework problem from Algebra by Hungerford.
Let $R$ be the following subring of the complex numbers: $R = \{a + b(1 + \sqrt{19}i)/2 \mid a, b \in \mathbb{Z}\}$. Then $R$ is a principal ideal domain that is not a Euclidean domain.
I've shown that it is not a Euclidean domain, but I can't seem to show that it is a principal ideal domain. There's only a few details that are bothering me.
Here is a proof outline that I've been given:
We will use the function $N(x) = x\bar{x}$, and notation $\theta = (1 + \sqrt{19}i)/2$. Let $I$ be an ideal in $R$, and choose $b \in I \backslash \{0\}$ which minimizes $N$ (i.e. $N(b) \le N(a)$ for all $a \in I \backslash \{0\}$).
- If $I \backslash Rb$ is non-empty, then there is $a \in I \backslash Rb$ with $\lvert Im(a/b) \rvert \le \sqrt{19}/4$ (of course $Im(x)$ is the imaginary part of $x$).
- If $\lvert Im(a/b) \rvert < \sqrt{3}/2$ then there is $m \in \mathbb{Z}$ with $N(a/b – m) < 1$.
- If $\sqrt{3}/2 \le \lvert Im(a/b) \rvert \le \sqrt{19}/4$ then $0 \le \lvert Im(2a/b-\theta) \rvert \le \sqrt{3} – \sqrt{19}/2$ [I think this should be $\sqrt{19}/2 – \sqrt{3}$]. Therefore there is an $m \in \mathbb{Z}$ with $N(2a/b – \theta – m) < 1$.
- Conclude that $I = Rb$.
The only part I'm having trouble with is part 1. Everything else is clear to me.
How can I find such an $a$?
Thanks for any help! And thanks to Gerry Myerson for clearing up part 3.
Best Answer
For first sentence of part 3, you have $${\sqrt3\over2}\le\Im(a/b)\le{\sqrt{19}\over4}{\rm\ or\ }-{\sqrt3\over2}\ge\Im(a/b)\ge-{\sqrt{19}\over4}$$ so $$\sqrt3\le\Im(2a/b)\le{\sqrt{19}\over2}{\rm\ or\ }-\sqrt3\ge\Im(2a/b)\ge-{\sqrt{19}\over2}$$ The imaginary part of $\theta$ is $\sqrt{19}/2$, so $$\sqrt3-{\sqrt{19}\over2}\le\Im((2a/b)-\theta)\le0{\rm\ or\ }-\sqrt3+{\sqrt{19}\over2}\ge\Im((2a/b)+\theta)\ge0$$ which is not exactly what you claim, but if you are willing to put up with $\pm\theta$ in place of $\theta$ it is good enough to get what you need.