scholarly journals Finite arithmetic subgroups of GLn, IV

1996 ◽  
Vol 142 ◽  
pp. 183-188 ◽  
Author(s):  
Yoshiyuki Kitaoka ◽  
Hiroshi Suzuki
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Number ◽  
Finite Subgroup ◽  
The Third ◽  
Finite Arithmetic

In this paper, we improve a result of the third paper of this series, that is we showTHEOREM. Let K be a nilpotent extension of the rational number field Q with Galois group Γ, and G a Γ- stable finite subgroup of GLn(0K). Then G is of A-type.

scholarly journals Finite arithmetic subgroups of GLn, II

1980 ◽  
Vol 77 ◽  
pp. 137-143 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Galois Extension ◽  
Finite Subgroup ◽  
Maximal Order ◽  
Finite Arithmetic ◽  
Totally Real

In [1] ∼ [6] the following question was treated: Let k be a totally real Galois extension of the rational number field Q, O the maximal order of k and G a finite subgroup of GL(n, O) which is stable under the operation of G(k/Q). Then does G ⊂ GL(n, Z) hold?

scholarly journals Finite arithmetic subgroups of GLn, V

1997 ◽  
Vol 146 ◽  
pp. 131-148 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Galois Extension ◽  
Finite Subgroup ◽  
Finite Arithmetic

Abstract.Let K be a finite Galois extension of the rational number field Q and G a Gal(K/Q)-stable finite subgroup of GLn(OK). We have shown that G is of A-type in several cases under some restrictions on K. In this paper, we show that it is true for n = 2 without any restrictions on K.

scholarly journals On Galois groups of class two extensions over the rational number field

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

On tamely ramified pro-p-extensions over -extensions of

2013 ◽  
Vol 156 (2) ◽  
pp. 281-294
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Rational Number ◽  
Prime Numbers ◽  
Finite Set

AbstractFor an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

scholarly journals ON THE ℤp-RANKS OF TAMELY RAMIFIED IWASAWA MODULES

2013 ◽  
Vol 09 (06) ◽  
pp. 1491-1503 ◽  
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA ◽  
MANABU OZAKI
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Number ◽  
Quadratic Field ◽  
Linear Independence ◽  
Prime Numbers ◽  
Imaginary Quadratic Field ◽  
Algebraic Numbers ◽  
Finite Set ◽  
Explicit Formulae

For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic ℤp-extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, we give the explicit formulae of the ℤp-ranks of the S-ramified Iwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.

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