2. The Genus Field and the Genus Group

1983 ◽  
pp. 11-28
Author(s):  
Albrecht Fröhlich
Keyword(s):  
Genus Field

scholarly journals On unramified cyclic extensions of degree l of algebraic number fields of degree l

1987 ◽  
Vol 107 ◽  
pp. 135-146 ◽  
Author(s):  
Yoshitaka Odai
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Preceding Paper ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Maximal Extension ◽  
Algebraic Number Fields ◽  
Cubic Field ◽  
Genus Field ◽  
Abelian Number Field

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

scholarly journals Abundant central extensions of non-trivial genera

1984 ◽  
Vol 95 ◽  
pp. 51-62 ◽  
Author(s):  
K. Miyake ◽  
N. Ormerod
Keyword(s):  
Galois Extension ◽  
Algebraic Closure ◽  
Abelian Extension ◽  
Global Field ◽  
Central Extensions ◽  
Genus Field

Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.

scholarly journals Some further remarks on genus field

1985 ◽  
Vol 21 (3) ◽  
pp. 256-259 ◽  
Author(s):  
M Bhaskaran
Keyword(s):  
Genus Field

scholarly journals Principal ideal theorems in the genus field for absolutely abelian extensions

1977 ◽  
Vol 9 (1) ◽  
pp. 4-15 ◽  
Author(s):  
Hisako Furuya
Keyword(s):  
Principal Ideal ◽  
Abelian Extensions ◽  
Genus Field

scholarly journals Cyanoboletus macroporus (Boletaceae), a new bolete species from Pakistani forests Samina

Karstenia ◽  
2021 ◽  
pp. 78-87
Author(s):  
Samina Sarwar ◽  
Arooj Naseer ◽  
Abdul N. Khalid
Keyword(s):  
Phylogenetic Analyses ◽  
Its Region ◽  
Morphological Data ◽  
Species Number ◽  
Closely Related Species ◽  
Distinctive Features ◽  
Line Drawings ◽  
Unknown Species ◽  
Genus Field ◽  
Comparison Table

<em>Cyanoboletus macroporus</em> belonging to <em>C. pulverulentus</em> species complex is designated as a new species from the moist temperate and sub-alpine oak forests of Pakistan after in depth macroscopic, microscopic and phylogenetic analyses using the ITS region of nrDNA as well as comparison with allied taxa. This species belonging to Boletoid group is morphologically distinguished from allied taxa (<em>Cyanoboletus flavosanguineus</em>, <em>C. hymenoglutinosus</em>, <em>C. pulverulentus</em>, <em>C. rainisii</em>, and <em>C. sinopulverulentus</em>) by wider openings of pores. <em>C. macroporus</em> is also phylogenetically distinct from <em>C. sinopulverulentus</em> and <em>C. pulverulentus</em>, the most closely related species. Phylogenetic analysis outlined the existence of previously unknown species of this genus. Field photographs of fresh basidocarps and line drawings of micro-characters are provided along with a phylogenetic tree as well as a comparison table and a key of distinctive features of all the species in this genus. This is the first authentic species belonging to <em>Cyanoboletus</em> from Pakistan. Previously, only <em>C. pulverulentus</em> has been mentioned in literature, but no morphological data is available regarding this report. With the addition of this taxon, species number of <em>Cyanoboletus</em> will increase to eight. From Pakistan, despite of the fact that there is great diversity of mushrooms in moist temperate areas (Yousaf et al. 2012), this is the first study that describes a species belonging to <em>Cyanoboletus</em> genus. Previously only one ambiguous species, <em>Cyanoboletus pulverulentus</em>, has been mentioned in literature (Iqbal & Khalid 1996), but with no available materials that could confirm this finding. In this study, <em>Cyanoboletus macroporus</em> is described as a new to science and increase the current species number of <em>Cyanoboletus</em> to eight.

scholarly journals The Factorization of $p$ in $\mathbf{Q}(a^{1/p^k})$ and the Genus Field of $\mathbf{Q}(a^{1/n})$

1988 ◽  
Vol 11 (1) ◽  
pp. 1-19 ◽  
Author(s):  
William Yslas VÉLEZ
Keyword(s):  
Genus Field

Genus fields of Kummer ℓn-cyclic extensions

2021 ◽  
pp. 2150062
Author(s):  
Carlos Daniel Reyes-Morales ◽  
Gabriel Villa-Salvador
Keyword(s):  
Prime Number ◽  
Function Field ◽  
Function Fields ◽  
Cyclic Extension ◽  
Abelian Extensions ◽  
Genus Field

We give a construction of the genus field for Kummer [Formula: see text]-cyclic extensions of rational congruence function fields, where [Formula: see text] is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer [Formula: see text]-cyclic extension. Finally, we study the extension [Formula: see text], for [Formula: see text], [Formula: see text] abelian extensions of [Formula: see text].

scholarly journals A principal ideal theorem in the genus field

1971 ◽  
Vol 23 (4) ◽  
pp. 697-718 ◽  
Author(s):  
Fumiyuki Terada
Keyword(s):  
Principal Ideal ◽  
Genus Field

scholarly journals ON THE STRONGLY AMBIGUOUS CLASSES OF 𝕜/ℚ(i) WHERE $\mathds{k} = {\mathbb Q}(\sqrt{2p_{1}p_{2}, i})$

2014 ◽  
Vol 07 (01) ◽  
pp. 1450021 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous
Keyword(s):  
Infinite Family ◽  
Number Fields ◽  
Class Groups ◽  
Quadratic Extensions ◽  
Genus Field ◽  
The Absolute

We construct an infinite family of imaginary bicyclic biquadratic number fields 𝕜 with the 2-ranks of their 2-class groups are ≥ 3, whose strongly ambiguous classes of 𝕜/ℚ(i) capitulate in the absolute genus field 𝕜(*), which is strictly included in the relative genus field (𝕜/ℚ(i))* and we study the capitulation of the 2-ideal classes of 𝕜 in its quadratic extensions included in 𝕜(*).

Sign in / Sign up

Export Citation Format

Share Document