A survey of computational class field theory

Henri Cohen

Displaying similar documents to “A survey of computational class field theory”

Class numbers of real cyclic number fields with small conductor

John Myron Masley (1978)

Compositio Mathematica

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The class number one problem for some non-abelian normal CM-fields of degree $24$

F. Lemmermeyer, S. Louboutin, R. Okazaki (1999)

Journal de théorie des nombres de Bordeaux

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We determine all the non-abelian normal CM-fields of degree 24 with class number one, provided that the Galois group of their maximal real subfields is isomorphic to $𝒜_{4}$ , the alternating group of degree $4$ and order $12$ . There are two such fields with Galois group $𝒜_{4} \times 𝒞_{2}$ (see Theorem 14) and at most one with Galois group SL $_{2} (𝔽_{3})$ (see Theorem 18); if the generalized Riemann hypothesis is true, then this last field has class number $1$ .

Some unsolved problems

W. Narkiewicz (1971)

Mémoires de la Société Mathématique de France

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Stickelberger elements in function fields

David R. Hayes (1985)

Compositio Mathematica

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CM-fields with all roots of unity

Kuniaki Horie (1990)

Compositio Mathematica

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On $ω_{1}$ -categorical theories of fields

Angus Macintyre (1971)

Fundamenta Mathematicae

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Ten problems on quadratic forms

Kazimierz Szymiczek (2002)

Acta Mathematica et Informatica Universitatis Ostraviensis

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The imaginary abelian number fields with class numbers equal to their genus class numbers

Ku-Young Chang, Soun-Hi Kwon (2000)

Journal de théorie des nombres de Bordeaux

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We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.

Some results on local fields

Akram Lbekkouri (2013)

Annales UMCS, Mathematica

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Let K be a local field with finite residue field of characteristic p. This paper is devoted to the study of the maximal abelian extension of K of exponent p−1 and its maximal p-abelian extension, especially the description of their Galois groups in solvable case. Then some properties of local fields in general case are studied too.