All Questions
4
questions
0
votes
1
answer
82
views
An example showing $\mathbb{Z}[\sqrt[3]{7}]$ is not a UFD [closed]
It cannot be a UFD because it's the ring of integers of $\mathbb{Q}(\sqrt[3]{7})$ and has class number 3. How can we give an example showing this?
- 55
3
votes
1
answer
293
views
Confusion about ideal class group computation
I am attempting to compute the ideal class group of the real quadratic field $K = \Bbb Q(\sqrt{65})$, which has ring of integers $\mathcal{O}_K = \Bbb Z\left[\frac{1 + \sqrt{65}}{2}\right]$.
The ...
- 4,608
1
vote
2
answers
233
views
Is there always a prime having only one prime above it?
Given a number field $K / \Bbb Q$, can we find a prime $p \in \Bbb Z$ which has only one prime of $K$ above it (e.g. $p$ is inert or $p$ is totally ramified)?
For instance, if $K$ is Galois over $\...
- 6,372
2
votes
0
answers
99
views
Lattice Bases of Prime (Ideal) Divisors
My question is:
How can I find the prime (ideal) divisors of 2 and 3 in the ring of integers of $\mathbb Q[\sqrt{-14}]$ and $\mathbb Q[\sqrt{-23}]$?
Here's what I have so far.
I found that (2, $\...
- 311