In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, that is then there exists in L such that The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht (Hilbert , ), although it is originally due to Kummer .
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| - Hilberts Satz 90 (de)
- Hilbert's Theorem 90 (en)
- Théorème 90 de Hilbert (fr)
- ヒルベルトの定理90 (ja)
- Теорема Гильберта 90 (ru)
- Теорема Гільберта 90 (uk)
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| - Der mathematische Satz, den David Hilbert unter der Nummer 90 in seiner Theorie der algebraischen Zahlkörper aufführt und der seither diesen Namen trägt, macht eine Aussage über die Struktur bestimmter Körpererweiterungen. Er wurde en passant bereits 1855 von Kummer bewiesen. (de)
- En théorie de Galois, le théorème 90 de Hilbert est une propriété algébrique d'énoncé simple et de grande portée par son interprétation homologique. Ce théorème tire son nom de l'ouvrage paru en 1897, (en), par David Hilbert, dans lequel il est énoncé, et démontré, comme théorème 90. Il a été ensuite généralisé par Emmy Noether. (fr)
- 数学、特に体論において、ヒルベルトの定理90 (Hilbert's Theorem 90) は、体の巡回拡大に関する重要な定理である。 (ja)
- Теорема Гільберта 90 — одне з основних тверджень для скінченних циклічних розширень Галуа E/K . (uk)
- Теоре́ма Ги́льберта 90 — одно из основных утверждений для конечных циклических расширений Галуа. (ru)
- In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, that is then there exists in L such that The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht (Hilbert , ), although it is originally due to Kummer . (en)
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| - Emmy Noether (en)
- Ernst Kummer (en)
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| - Der mathematische Satz, den David Hilbert unter der Nummer 90 in seiner Theorie der algebraischen Zahlkörper aufführt und der seither diesen Namen trägt, macht eine Aussage über die Struktur bestimmter Körpererweiterungen. Er wurde en passant bereits 1855 von Kummer bewiesen. (de)
- In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, that is then there exists in L such that The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht (Hilbert , ), although it is originally due to Kummer . Often a more general theorem due to Emmy Noether is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group of G, with coefficients in the multiplicative group of L, is trivial: (en)
- En théorie de Galois, le théorème 90 de Hilbert est une propriété algébrique d'énoncé simple et de grande portée par son interprétation homologique. Ce théorème tire son nom de l'ouvrage paru en 1897, (en), par David Hilbert, dans lequel il est énoncé, et démontré, comme théorème 90. Il a été ensuite généralisé par Emmy Noether. (fr)
- 数学、特に体論において、ヒルベルトの定理90 (Hilbert's Theorem 90) は、体の巡回拡大に関する重要な定理である。 (ja)
- Теорема Гільберта 90 — одне з основних тверджень для скінченних циклічних розширень Галуа E/K . (uk)
- Теоре́ма Ги́льберта 90 — одно из основных утверждений для конечных циклических расширений Галуа. (ru)
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