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

<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.

Genus fields of Kummer ℓn-cyclic extensions

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].

Washington units, semispecial units, and annihilation of class groups

Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called “semispecial”). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly weaker statement holds true for all abelian fields. The group of Washington units is very often larger than Sinnott’s group of cyclotomic units. In a companion paper we will show that in concrete families of abelian fields the group of Washington units is much larger than that of Sinnott units, by giving lower bounds on the index. Combining this with the present paper gives strong annihilation results.

On the capitulation of the $2$-ideal classes of the field Q(\sqrt{pq_1q_2}, i) of type (2, 2, 2)

We study the capitulation of the 2-ideal classes of the field k =Q(\sqrt{p_1p_2q}, \sqrt{-1}), where p_1\equiv p_2\equiv-q\equiv1 \pmod 4 are different primes, in its three quadratic extensions contained in its absolute genus field k^{*} whenever the 2-class group of $\kk$ is of type $(2, 2, 2)$.

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

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(*).

GENUS FIELDS OF CYCLIC l-EXTENSIONS OF RATIONAL FUNCTION FIELDS

We give a construction of genus fields for Kummer cyclic l-extensions of rational congruence function fields, l a prime number. First we find this genus field for a field contained in a cyclotomic function field using Leopoldt's construction by means of Dirichlet characters and the Hilbert class field defined by Rosen. The general case follows from this. This generalizes the result obtained by Peng for a cyclic extension of degree l.

RÉDEI FIELDS AND DYADIC EXTENSIONS

For an arbitrary non-square discriminant D, the Rédei fieldΓ0(D) is introduced as an extension of [Formula: see text] analogous to the genus field and connected with the Rédei–Reichardt Theorem. It is shown how to compute Rédei fields, and this is used to find socles of dyadic extensions of K for negative D. Finally, a theorem and two conjectures are presented relating the fields [Formula: see text] and [Formula: see text] for an odd prime p.

On the rank of the first radical layer of a p-class group of an algebraic number field

Let p be a prime number. Let M be a finite Galois extension of a finite algebraic number field k. Suppose that M contains a primitive pth root of unity and that the p-Sylow subgroup of the Galois group G = Gal(M/k) is normal. Let K be the intermediate field corresponding to the p-Sylow subgroup. Let = Gal(K/k). The p-class group C of M is a module over the group ring ZpG, where Zp is the ring of p-adic integers. Let J be the Jacobson radical of ZpG. C/JC is a module over a semisimple artinian ring Fp. We study multiplicity of an irreducible representation Φ apperaring in C/JC and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field M such that the minus part of C is cyclic as a ZpG-module and a CM-field M such that the plus part of C vanishes for odd p.To show the formula, we apply theory of central extensions of algebraic number field and study global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to M/K.