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## 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.

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.