When is integral closure generated by one element

  • #1
187
9

Main Question or Discussion Point

Hello,
This is not a homework problem, nor a textbook question. Please do not remove.
Is there a concrete example of the following setup :
[itex]R[/itex] is an integrally closed domain,
[itex]a[/itex] is an integral element over [itex]R[/itex],
[itex]S[/itex] is the integral closure of [itex]R[a][/itex] in its fraction field,
[itex]S[/itex] is not of the form [itex]R{[}b{]}[/itex] for any element [itex]b[/itex] in [itex]S[/itex].

For example, the integral closure of [itex]{\mathbb Z}(\sqrt 5)[/itex] is the set of elements of the form [itex](m + \sqrt 5 n)/2[/itex], where [itex]m^2 - 5n^2[/itex] is a multiple of 4. So, [itex]S[/itex] is generated by [itex]\sqrt 5/2[/itex] over [itex]\mathbb Z[/itex]: This does not fulfill the desired conditions.
 

Answers and Replies

  • #2
mathwonk
Science Advisor
Homework Helper
10,803
972
i don't understand. in your example apparently R is not integrally closed. in fact under your conditions, that R is integrally closed and a is integral over R, this implies that R = R[a], doesn't it?

Oh sorry, you meant integral over it but in some finite field extension of its fraction field. so your R was Z and your a was sqrt(5). got it. and maybe you meant Z[sqrt(5)] instead of Z(sqrt(5))?
 
Last edited:
  • #3
187
9
Yes, I meant [itex]{\mathbb Z}{[}\sqrt 5{]}.[/itex]
In my example, [itex]R = {\mathbb Z},\ a = \sqrt 5[/itex] and [itex]S[/itex] is the integral closure of [itex]{\mathbb Z}[\sqrt 5][/itex] in [itex]{\mathbb Z}(\sqrt 5)[/itex], which has the form given above.
 
  • #4
mathwonk
Science Advisor
Homework Helper
10,803
972
the answer is there does exist such a counter example Z[a] to the integral closure equaling Z[ b]. all your examples are numbers fields, and all number fields fall into your examples. You are asking for the structure of the ring of integers, which is known to have a finite basis as an abelian group. You want that basis to be the powers, up to some finite power, of a single integral element.
It is well known since Richard Dedekind that not all rings of integers in number fields have so called "power bases". here is a reference: (or just search on "non monogenic fields".)

http://wstein.org/129-05/final_papers/Yan_Zhang.pdf
 
Last edited:
  • #5
187
9
Very useful. I was completely unaware of this topic. Thank you so many mathwonk.
 

Related Threads for: When is integral closure generated by one element

Replies
13
Views
3K
Replies
6
Views
825
  • Last Post
Replies
3
Views
626
  • Last Post
Replies
22
Views
7K
  • Last Post
Replies
20
Views
6K
Replies
6
Views
536
Top