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| :''This article is about the Hauptidealsatz of [[class field theory]]. You may be seeking [[Krull's principal ideal theorem]], also known as Krull's Hauptidealsatz, in [[commutative algebra]]''
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| In [[mathematics]], the '''principal ideal theorem''' of [[class field theory]], a branch of [[algebraic number theory]], is the statement that for any [[algebraic number field]] ''K'' and any ideal ''I'' of the [[ring of integers]] of ''K'', if ''L'' is the [[Hilbert class field]] of ''K'', then
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| :<math>IO_L\ </math> | |
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| is a [[principal ideal]] α''O''<sub>''L''</sub>, for ''O''<sub>''L''</sub> the ring of integers of ''L'' and some element α in it. In other terms, extending ideals gives a mapping on the [[class group]] of ''K'', to the class group of ''L'', which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called ''principalization'', or sometimes ''capitulation''. It was conjectured by [[David Hilbert]], and was the last remaining aspect of his programme on class fields to be completed, around 1930.
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| The question was reduced to a piece of finite group theory by [[Emil Artin]]. That involved the [[transfer (group theory)|transfer]]. The required result was proved by [[Philipp Furtwängler]].
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| ==References==
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| * {{cite journal | first=Philipp | last=Furtwängler | authorlink=Philipp Furtwängler | title=Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper | journal=Abh. Math. Sem. Hamburg | volume=7 | year=1929 | pages=14–36 | jfm=55.0699.02 }}
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| * {{cite book | last=Gras | first=Georges | title=Class field theory. From theory to practice | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=2003 | isbn=3-540-44133-6 | zbl=1019.11032 }}
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| * {{cite book | first=Helmut | last=Koch | title=Algebraic Number Theory | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-63003-1 | zbl=0819.11044 | series=Encycl. Math. Sci. | volume=62 | edition=2nd printing of 1st | page=104 }}
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| *{{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local fields | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | pages=120–122 }}
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| [[Category:Ideals]]
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| [[Category:Group theory]]
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| [[Category:Homological algebra]]
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| [[Category:Theorems in algebraic number theory]]
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