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- Principalization_(algebra) 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 it's 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 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.".
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- Principalization_(algebra) wikiPageOutDegree "89".
- Principalization_(algebra) wikiPageRevisionID "682681870".
- Principalization_(algebra) wikiPageWikiLink Abelian_extension.
- Principalization_(algebra) wikiPageWikiLink Algebraic_number_field.
- Principalization_(algebra) wikiPageWikiLink Algebraic_number_theory.
- Principalization_(algebra) wikiPageWikiLink Arnold_Scholz.
- Principalization_(algebra) wikiPageWikiLink Artin_reciprocity_law.
- Principalization_(algebra) wikiPageWikiLink Artin_transfer_(group_theory).
- Principalization_(algebra) wikiPageWikiLink Biquadratic_field.
- Principalization_(algebra) wikiPageWikiLink Category:Class_field_theory.
- Principalization_(algebra) wikiPageWikiLink Category:Group_theory.
- Principalization_(algebra) wikiPageWikiLink Class_field.
- Principalization_(algebra) wikiPageWikiLink Class_field_theory.
- Principalization_(algebra) wikiPageWikiLink Commutator_subgroup.
- Principalization_(algebra) wikiPageWikiLink Conductor_(class_field_theory).
- Principalization_(algebra) wikiPageWikiLink Conjugacy_class.
- Principalization_(algebra) wikiPageWikiLink Cubic_field.
- Principalization_(algebra) wikiPageWikiLink David_Hilbert.
- Principalization_(algebra) wikiPageWikiLink Degree_of_a_field_extension.
- Principalization_(algebra) wikiPageWikiLink Descendant_tree_(group_theory).
- Principalization_(algebra) wikiPageWikiLink Discriminant_of_an_algebraic_number_field.
- Principalization_(algebra) wikiPageWikiLink Double_coset.
- Principalization_(algebra) wikiPageWikiLink Emil_Artin.
- Principalization_(algebra) wikiPageWikiLink Equivariant_map.
- Principalization_(algebra) wikiPageWikiLink Ernst_Kummer.
- Principalization_(algebra) wikiPageWikiLink Extension_and_contraction_of_ideals.
- Principalization_(algebra) wikiPageWikiLink Field_extension.
- Principalization_(algebra) wikiPageWikiLink Field_norm.
- Principalization_(algebra) wikiPageWikiLink File:ClassExtensionAndArtinTransfer.png.
- Principalization_(algebra) wikiPageWikiLink Fractional_ideal.
- Principalization_(algebra) wikiPageWikiLink Frobenius_endomorphism.
- Principalization_(algebra) wikiPageWikiLink G-set.
- Principalization_(algebra) wikiPageWikiLink Galois_cohomology.
- Principalization_(algebra) wikiPageWikiLink Galois_extension.
- Principalization_(algebra) wikiPageWikiLink Galois_group.
- Principalization_(algebra) wikiPageWikiLink Galois_module.
- Principalization_(algebra) wikiPageWikiLink Group_action.
- Principalization_(algebra) wikiPageWikiLink Group_extension.
- Principalization_(algebra) wikiPageWikiLink Group_homomorphism.
- Principalization_(algebra) wikiPageWikiLink Group_theory.
- Principalization_(algebra) wikiPageWikiLink Herbrand_quotient.
- Principalization_(algebra) wikiPageWikiLink Hilbert_class_field.
- Principalization_(algebra) wikiPageWikiLink Hilberts_Theorem_90.
- Principalization_(algebra) wikiPageWikiLink Ideal_(ring_theory).
- Principalization_(algebra) wikiPageWikiLink Ideal_class_group.
- Principalization_(algebra) wikiPageWikiLink Ideal_norm.
- Principalization_(algebra) wikiPageWikiLink Ideal_number.
- Principalization_(algebra) wikiPageWikiLink Kernel_(algebra).
- Principalization_(algebra) wikiPageWikiLink Maximal_abelian_extension.
- Principalization_(algebra) wikiPageWikiLink Metabelian_group.
- Principalization_(algebra) wikiPageWikiLink Modulus_(algebraic_number_theory).
- Principalization_(algebra) wikiPageWikiLink Non-abelian_group.
- Principalization_(algebra) wikiPageWikiLink Normal_extension.
- Principalization_(algebra) wikiPageWikiLink Number_theory.
- Principalization_(algebra) wikiPageWikiLink Olga_Taussky.
- Principalization_(algebra) wikiPageWikiLink Olga_Taussky-Todd.
- Principalization_(algebra) wikiPageWikiLink Orbit-stabilizer_theorem.
- Principalization_(algebra) wikiPageWikiLink P-group_generation_algorithm.
- Principalization_(algebra) wikiPageWikiLink Philipp_Furtwängler.
- Principalization_(algebra) wikiPageWikiLink Prime_ideal.
- Principalization_(algebra) wikiPageWikiLink Principal_ideal.
- Principalization_(algebra) wikiPageWikiLink Principal_ideal_theorem.
- Principalization_(algebra) wikiPageWikiLink Pro-p_group.
- Principalization_(algebra) wikiPageWikiLink Quadratic_field.
- Principalization_(algebra) wikiPageWikiLink Ramification_(mathematics).
- Principalization_(algebra) wikiPageWikiLink Ring_of_integers.
- Principalization_(algebra) wikiPageWikiLink Splitting_of_prime_ideals_in_Galois_extensions.
- Principalization_(algebra) wikiPageWikiLink Springer-Verlag.
- Principalization_(algebra) wikiPageWikiLink Springer_Science+Business_Media.
- Principalization_(algebra) wikiPageWikiLink Takagi_existence_theorem.
- Principalization_(algebra) wikiPageWikiLink Tate_cohomology_group.
- Principalization_(algebra) wikiPageWikiLink Unit_(ring_theory).
- Principalization_(algebra) wikiPageWikiLink Unramified_extension.
- Principalization_(algebra) wikiPageWikiLink Zahlbericht.
- Principalization_(algebra) wikiPageWikiLinkText "''principalization''".
- Principalization_(algebra) wikiPageWikiLinkText "Hilbert ''p''-class field towers".
- Principalization_(algebra) wikiPageWikiLinkText "class extension homomorphisms".
- Principalization_(algebra) wikiPageWikiLinkText "principalization problem".
- Principalization_(algebra) hasPhotoCollection Principalization_(algebra).
- Principalization_(algebra) wikiPageUsesTemplate Template:Cite_book.
- Principalization_(algebra) wikiPageUsesTemplate Template:Reflist.
- Principalization_(algebra) subject Category:Class_field_theory.
- Principalization_(algebra) subject Category:Group_theory.
- Principalization_(algebra) 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 it's extension to the ring of integers of the larger field is.".
- Principalization_(algebra) label "Principalization (algebra)".
- Principalization_(algebra) sameAs m.010xd0zl.
- Principalization_(algebra) sameAs Q17300011.
- Principalization_(algebra) sameAs Q17300011.
- Principalization_(algebra) wasDerivedFrom Principalization_(algebra)?oldid=682681870.
- Principalization_(algebra) isPrimaryTopicOf Principalization_(algebra).