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Published: 12 septembre 2022 (2 semaines ago)



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Showing the ring of integers is a Dedekind domain

Let $K$ be a number field and $O_K$ be the ring of integers of $K$. I want to show that $O_K$ is a Dedekind domain.
I saw this theorem in Wikipedia,
If $K$ is a number field and $O_K$ is the ring of integers of $K$, then
a) $O_K$ is a Dedekind domain.
b) $O_K$ is a UFD.
c) The ring of integers of a number field is a Dedekind domain.
I am not sure about how to prove this theorem.
Please help me prove this theorem, thanks in advance.


Let $L$ be the number field $mathbb{Q}(sqrt[4]{5})$. $L$ is a real quadratic extension of $mathbb{Q}$ and $O_L$ is a Dedekind domain. Therefore every $mathbb{Z}$-submodule of $O_L$ containing $1$ has a $mathbb{Z}$-generator. Now $O_K$ contains $O_L$, therefore $O_K$ contains $1$, hence $O_K$ is a Dedekind domain.


Let $A=mathbb Z cap O_K$ be the intersection of the ring of integers and the ring of integers. It’s a $mathbb Z$-module containing $1$ and with non zero divisors. By Hilbert’s theorem 90, the group $A$ is free so it’s free as a $mathbb Z$-module and this implies that it’s projective (by Nakayama’s lemma) so that $text{fd}_{mathbb Z}A=text{fd}_{mathbb Z}mathbb Z=0$.


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Mirai virus is a powerful ransomware virus. It encrypts all type of files and asks for a ransom of $250, not so difficult to pay. McAfee researchers reported a new variant of the « Mirai » virus.With the deaths of two B.C. infants and a toddler in Darke County, some are calling for an investigation into whether the deaths were the result of an “overall lack of safety” in the community’s health care system.

Early Sunday morning, 1-year-old Caladenia (Cali) Middelkotte was found dead of what are believed to be non-fatal head injuries in the 14700 block of West Marion Road in rural Darke County. According to a statement released by Darke County Coroner Sherry McNutt, emergency workers found Cali “unresponsive with multiple injuries, and eventually pronounced dead at the scene.” The coroner, who is still trying to determine how the toddler died, said the injuries were “inflicted,” but that it’s too early in the investigation to determine cause or manner of death.

The full statement, obtained by CTV News, reads as follows:

“On Sunday October 17, 2014 at 1:07 a.m. CDT, the Darke County Coroner’s Office received a 911 call in reference to a one-year-old boy (fetus) whose body was found in the 14700 block of West Marion Road. Emergency Medical Services (EMS) and law enforcement responded to the scene. The child was in a lying position in a dry creek bed and was pronounced deceased at the scene.

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On Tuesday, 13-month-old Zephaniah Middelkotte was found dead in the same area. The boy was a toddler who was a patient at St. Anne’s Hospital in Marklesburg, more than 100 kilometres from Marion.

Zephaniah’s mother was driving home to the United States when she was pulled over on