https://artofproblemsolving.com/wiki/index.php?title=Integral_closure&feed=atom&action=historyIntegral closure - Revision history2024-03-28T08:19:49ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=Integral_closure&diff=27881&oldid=prevJam: Add category2008-09-09T23:02:37Z<p>Add category</p>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form a subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form a subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.</div></td></tr>
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</table>Jamhttps://artofproblemsolving.com/wiki/index.php?title=Integral_closure&diff=7550&oldid=prevDVO: s --> a2006-07-13T09:09:49Z<p>s --> a</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 09:09, 13 July 2006</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form <del class="diffchange diffchange-inline">s </del>subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form <ins class="diffchange diffchange-inline">a </ins>subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.</div></td></tr>
</table>DVOhttps://artofproblemsolving.com/wiki/index.php?title=Integral_closure&diff=7540&oldid=prevComplexZeta at 02:11, 13 July 20062006-07-13T02:11:33Z<p></p>
<p><b>New page</b></p><div>Let <math>S</math> be a [[ring]] and <math>R</math> a subring of <math>S</math>. We say that an element <math>s\in S</math> is '''integral''' over <math>R</math> if there is a [[monic polynomial]] <math>f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0</math>, with each <math>a_i\in R</math> so that <math>f(s)=0</math>. The elements of <math>S</math> that are integral over <math>R</math> form s subring <math>T</math> of <math>S</math> which contains <math>R</math>. We call <math>T</math> the '''integral closure''' of <math>R</math> in <math>S</math>. If <math>T=S</math>, then we say that <math>S</math> is '''integral''' over <math>R</math>. If <math>T=R</math>, then we say that <math>R</math> is '''integrally closed''' in <math>S</math>. If <math>R</math> is integrally closed in its [[field of fractions]], then we call it '''integrally closed'''.</div>ComplexZeta