Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

When is integral closure generated by one element

  1. Feb 5, 2015 #1
    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.
     
  2. jcsd
  3. Feb 5, 2015 #2

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    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: Feb 5, 2015
  4. Feb 5, 2015 #3
    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.
     
  5. Feb 5, 2015 #4

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    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: Feb 5, 2015
  6. Feb 6, 2015 #5
    Very useful. I was completely unaware of this topic. Thank you so many mathwonk.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: When is integral closure generated by one element
  1. Integral closure (Replies: 20)

Loading...