A Problem from Marcus' Number Fields

Asked Apr 19 '16 at 17:23

Active Apr 21 '16 at 06:36

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I have been stuck on the 17th problem of the 3rd chapter from Marcus' Number Fields.

Let $K=\mathbb{Q}[\sqrt-23]$ , $L=\mathbb{Q}[\omega]$ where $\omega = e^{2.\pi.i/23} $ . Let $P$ be one of the primes of $R= \mathbb{A} \cap K$ lying over $2$ ; secifically take $P=(2,\theta)$ where $\theta=(1+ \sqrt-23)/2$. Let $Q$ be a prime of $\mathbb{Z}[\omega]$ lying over $P$.Show that

$(a)$ $Q$ is not principal.

$(b)$ If $2=\alpha\beta$ with $\alpha,\beta \in \mathbb{Z}[\omega]$, then $\alpha$ or $\beta$ is unit in $\mathbb{Z}[\omega]$.

I just follow the hints given there.I know that ideal class group of $\mathbb{Z}[\omega]$ is isomorphic to $\mathbb{Z}_3$. But I can't go further. Need some help.

edited Apr 21 '16 at 06:36

asked Apr 19 '16 at 17:23

Saikat

For $(a)$: show that $P$ is not principal in the way it is shown in this question. – Dietrich Burde Apr 19 '16 at 17:40
Here we need to show that $Q$ is not principal – Saikat Apr 19 '16 at 17:48
This follows form the fact that $P$ is not principal, and $P^3=(\theta-2)$. – Dietrich Burde Apr 19 '16 at 17:49
1

How we can conclude that $Q$ is not principal from these two facts? – Saikat Apr 19 '16 at 17:53
Like Marcus says: use $16b.$ – Dietrich Burde Apr 19 '16 at 18:08
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Please elaborate these things..I didn't understand. – Saikat Apr 19 '16 at 18:13

A Problem from Marcus' Number Fields

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