Hilbert class field
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K. In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
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Algebraic number fieldAlgebraic number theoryClass field theoryClass formationComplex multiplicationDavid HilbertDiscriminant of an algebraic number fieldEvgeny GolodGenus fieldGolod–Shafarevich theoremGreenberg's conjecturesHilbert's twelfth problemHilbert fieldIdeal class groupIdeal numberList of algebraic number theory topicsList of things named after David HilbertLubin–Tate formal group lawNarrow Hilbert class fieldPlastic numberPrincipal idealPrincipal ideal theoremPrincipalization (algebra)Ray class fieldRing class fieldSiegel zeroStark conjecturesTakagi existence theoremTower of fields
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Hilbert class field
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K. In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
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En théorie algébrique des nomb ...... métique et théorie des nombres
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In algebraic number theory, th ...... to a complex embedding of E).
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代数的整数論において,数体 K のヒルベルト類体(英: Hi ...... 込みは E の実埋め込み(複素埋め込みではなく)に拡張する.
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David Hilbert
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David
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Hilbert
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Existence of Hilbert class field
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En théorie algébrique des nomb ...... lier une extension finie de K.
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In algebraic number theory, th ...... to a complex embedding of E).
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代数的整数論において,数体 K のヒルベルト類体(英: Hi ...... 込みは E の実埋め込み(複素埋め込みではなく)に拡張する.
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Corps de classes de Hilbert
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Hilbert class field
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ヒルベルト類体
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