The Galois module structure of algebraic integer rings in fields with generalised quaternion group

A. Fröhlich

Displaying similar documents to “The Galois module structure of algebraic integer rings in fields with generalised quaternion group”

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 $1$ .

Galois module structure of rings of integers

Martin J. Taylor (1980)

Annales de l'institut Fourier

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Let $E / F$ be a Galois extension of number fields with $Γ =$ Gal $(E / F)$ and with property that the divisors of $(E : F)$ are non-ramified in $E / Q$ . We denote the ring of integers of $E$ by $𝒪_{E}$ and we study $𝒪_{E}$ as a $Z Γ$ -module. In particular we show that the fourth power of the (locally free) class of $𝒪_{E}$ is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of $ℰ_{E}$ , together with new determinantal congruences for cyclic group rings and corresponding...

On Iwasawa’s λ-invariant for certain $Z_{l}$ -extensions

Joseph Carroll, H. Kisilevsky (1981)

Acta Arithmetica

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CM-fields with all roots of unity

Kuniaki Horie (1990)

Compositio Mathematica

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On Gauss sum characters of finite groups and generalized Bernoulli numbers

Shoichi Nakajima (1995)

Journal de théorie des nombres de Bordeaux

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Brauer's class number relation

C. Walter (1979)

Acta Arithmetica

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Tamely ramified extensions's structure.

Ibadula, Denis (2001)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

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Henselian Discrete Valued Fields Admitting One-Dimensional Local Class Field Theory

Chipchakov, I. (2004)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: 11S31 12E15 12F10 12J20. This paper gives a characterization of Henselian discrete valued fields whose finite abelian extensions are uniquely determined by their norm groups and related essentially in the same way as in the classical local class field theory. It determines the structure of the Brauer groups and character groups of Henselian discrete valued strictly primary quasilocal (or PQL-) fields, and thereby, describes the forms...

Kida's formula and congruences.

Pollack, Robert, Weston, Tom (2007)

Documenta Mathematica

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Class groups of abelian fields, and the main conjecture

Cornelius Greither (1992)

Annales de l'institut Fourier

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This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $p = 2$ , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $χ$ -parts of $p$ -class groups of abelian number fields: first for relative class groups of real fields (again including the case $p = 2$ ). As a consequence, a generalization of the Gras conjecture...