# Zulip Chat Archive

## Stream: FLT-regular

### Topic: Completion of FLT-regular

#### Riccardo Brasca (Dec 05 2023 at 12:57):

Let's use this topic to discuss future work or whatever.

#### Riccardo Brasca (Dec 05 2023 at 13:06):

@Xavier Roblot How far are from using Minkowski's bound to prove that $\mathbb{Q}(\zeta_5)$ is a PID? The bound is `1`

...

#### Riccardo Brasca (Dec 05 2023 at 13:38):

It seems that $\mathbb{Q}(\zeta_p)$ is norm euclidean for $p = 3, 5, 7, 11$, so maybe it is not completely impossible to prove that $p$ is regular in these cases.

(It is euclidean also if $p = 17, 19$ but I don't know if it is norm euclidean, and this is the complete list of primes such that is $\mathbb{Q}(\zeta_p)$ is a PID).

#### Johan Commelin (Dec 05 2023 at 14:30):

Huge congrats to all people involved!

#### Xavier Roblot (Dec 05 2023 at 14:42):

Riccardo Brasca said:

Xavier Roblot How far are from using Minkowski's bound to prove that $\mathbb{Q}(\zeta_5)$ is a PID? The bound is

`1`

...

I'm still fighting with the proof of Hermite's theorem at the moment but adapting the existing results in `CanonicalEmbedding`

to deduce the classical bound on the norm of ideals in ideal classes is next on my list. This would be enough to deduce that $\mathbb{Q}(\zeta_5)$ is principal. I am not sure about the other ones though (I'll have to do the computations when I have time).

#### Xavier Roblot (Dec 05 2023 at 14:42):

(I am in class now :grinning_face_with_smiling_eyes: )

#### Riccardo Brasca (Dec 05 2023 at 14:46):

For $p=7$ it is 8. Anyway getting $3$ and $5$ is already interesting.

#### Xavier Roblot (Dec 05 2023 at 14:49):

Well, we can still prove that all integral ideals of norm $\le 8$ are principal

Last updated: Dec 20 2023 at 11:08 UTC