About: Principal ideal theorem

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In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was

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rdfs:label	Principalization (algebra) (en)
rdfs:comment	In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was (en)
foaf:depiction
dcterms:subject	Group theory Class field theory
Wikipage page ID	43103430 (xsd:integer)
Wikipage revision ID	1064754309 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime ideal Principal ideal Quadratic field Modulus (algebraic number theory) Pro-p group David Hilbert Algebraic number field Ring of integers Cubic field Unit (ring theory) Degree of a field extension Descendant tree (group theory) Double coset Metabelian group Group theory Non-abelian group Emil Artin Frobenius endomorphism Conductor (class field theory) Conjugacy class Equivariant map Arnold Scholz Commutator subgroup Ideal (ring theory) Ideal class group Ideal norm Ideal number Kernel (algebra) Galois cohomology Galois extension Galois group Galois module Algebraic number theory Ernst Kummer Normal extension Number theory Discriminant of an algebraic number field Fractional ideal Hilbert's Theorem 90 Hilbert class field Group homomorphism Tate cohomology group Artin transfer (group theory) Abelian extension Class field theory Biquadratic field Herbrand quotient Principal ideal theorem Extension and contraction of ideals Artin reciprocity law Philipp Furtwängler Class field theory Degree of a field extension Field norm Group extension Group theory Maximal abelian extension Ramification (mathematics) Zahlbericht Splitting of prime ideals in Galois extensions Takagi existence theorem P-group generation algorithm Unramified extension Springer-Verlag Orbit-stabilizer theorem G-set Class field Olga Taussky
sameAs	Principal ideal theorem Principal ideal theorem Principal ideal theorem Principal ideal theorem
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thumbnail	wiki-commons:Special:FilePath/ClassExtensionAndArtinTransfer.png?width=300
has abstract	In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94. (en)
prov:wasDerivedFrom	wikipedia-en:Principalization_(algebra)?oldid=1064754309&ns=0
page length (characters) of wiki page	48453 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Principalization_(algebra)
is Link from a Wikipage to another Wikipage of	Capitulation (algebra) Ernst Kummer Artin transfer (group theory) Principal ideal theorem Artin reciprocity law
is Wikipage redirect of	Capitulation (algebra)
is foaf:primaryTopic of	wikipedia-en:Principalization_(algebra)

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