Hello,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums - The Fusion of Science and Community**

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

Tags:

Loading...

Similar Threads - integral closure generated | Date |
---|---|

I Finite Integral Domains ... Adkins & Weintraub, Propn 1.5 | Mar 11, 2018 |

I Irreducibles and Primes in Integral Domains ... | Apr 5, 2017 |

I Definition of an irreducible element in an integral domain | Feb 18, 2017 |

Quotient field of the integral closure of a ring | Dec 11, 2014 |

Integral closure | Oct 21, 2010 |

**Physics Forums - The Fusion of Science and Community**