I am self-studying a set of legitimately downloaded notes on algebraic number theory. They are somewhat akin to "Ireland and Rosen," Ch. 12.
I would appreciate help in understanding a proof (in the direction) that if $Cl_F$ consists of one equivalence class, then $D_F$, the ring of integers of $F$, is a PID. $F$ is a number field.
For convenience, I've numbered the four questions I would like help with.
Using the definition of equivalence in this context, if there is only one equivalence class, any two ideals of $D_F$ are equivalent and there exist $\alpha, \beta \in D_F$ such that
$\alpha I = \beta D_F$,
where $I$ is any ideal in $D_F$. We want to show $I$ is a principal ideal. So far so good. (1) Although the notes has $\alpha, \beta \in I$, I think they ought to be in $D_F$. Is this correct?
Thus $\alpha I = (\beta)$.
The proof goes on: Let $\omega = \beta \alpha^{- 1} \in F$.
Now I get stuck:
Then (2) $\omega I = \beta I \subseteq I$. I don't see how to get the equality.
Based on this, we know $\omega \in D_F$. This is clear since this type of assertion, based on the inclusion, was previously proved.
Then I also can't get the final assertion, (3) that this implies $(\omega) = I$.
Lastly, I would appreciate any guidance or references as to (4) in general what the product, e.g., $\gamma J$ means, where $J$ is an ideal.
As always, thanks for your patience and help.
