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Reference.org
Principal ideal theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the principal ideal theorem of <a href="/facts/Class_field_theory/4cvKbn6a">class field theory</a>, a branch of <a href="/facts/Algebraic_number_theory/LPzDT5CA">algebraic number theory</a>, says that extending ideals gives a mapping on the <a href="/facts/Class_group/LLn5Pwj1">class group</a> of an <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number field</a> to the class group of its <a href="/facts/Hilbert_class_field/UbGUNu89">Hilbert class field</a>, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called <a href="/facts/Principalization_(algebra)/NFhwhClY"><i>principalization</i></a>, or sometimes <i>capitulation</i>. </p>