The software package KANT V4 for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.

The nitrogen-fixing cyanobacterium Nostoc is a commonly occurring terrestrial and aquatic cyanobacterium often found in symbiosis with a wide range of plant, algal, and fungal species. We investigated the diversity of cyanobacterial species occurring within the coralloid roots of different Macrozamia cycad species at diverse locations throughout Australia. In all, 74 coralloid root samples were processed and 56 endosymbiotic cyanobacteria were cultured. DNA was isolated from unialgal cultures and a segment of the 16S rRNA gene was amplified and sequenced. Microscopic analysis was performed on representative isolates. Twenty-two cyanobacterial species were identified, comprising mostly Nostoc spp. and a Calothrix sp. No correlation was observed between a cycad species and its resident cyanobiont species. The predominant cyanobacterium isolated from 18 root samples occurred over a diverse range of environmental conditions and within 14 different Macrozamia spp. Phylogenetic analysis indicated that endosymbionts were not restricted to previously described terrestrial species. An isolate clustering with Nostoc PCC7120, an aquatic strain, was identified. This is the first comprehensive study to identify the endosymbionts within a cycad genus using samples obtained from their natural habitats. These results indicate that there is negligible host specialization of cyanobacterial endosymbionts within the cycad genus Macrozamia in the wild.

We describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.

Abstract. Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. We conclude with several examples. PreliminariesLet k/Q be an algebraic number field; we denote its ring of integers by o k . A congruence module m formally consists of an integral ideal m 0 and a formal product m ∞ of real places (.) (i) of k viewed as embeddings into R. By I m we denote the set of all ideals coprime to m 0 , by P m the set of principal ideals generated by elements α ≡ 1 mod * m (i.e., α ≡ 1 mod m 0 and α (i) > 0 for all (.) (i) |m ∞ ) and finally by Cl m := I m /P m the ray class group modulo m. There are efficient algorithms for computing ray class groups [6, 17] provided we already know the unit and class group of k.An ideal group H (defined mod m) is a subgroup of I m containing P m . For any ideal group H letH := I m /H.

The nitrogen-fixing bacterium, Nostoc, is a commonly occurring cyanobacterium often found in symbiotic associations. We investigated the potential of cycad cyanobacterial endosymbionts to synthesize microcystin/nodularin. Endosymbiont DNA was screened for the aminotransferase domain of the toxin biosynthesis gene clusters. Five endosymbionts carrying the gene were screened for bioactivity. Extracts of two isolates inhibited protein phosphatase 2A and were further analyzed using electrospray ionization mass spectrometry (ESI-MS)/MS. Nostoc sp. ‘Macrozamia riedlei 65.1’ and Nostoc sp. ‘Macrozamia serpentina 73.1’ both contained nodularin. High performance liquid chromatography (HPLC) HESI-MS/MS analysis confirmed the presence of nodularin at 9.55±2.4 ng μg−1 chlorophyll a in Nostoc sp. ‘Macrozamia riedlei 65.1’ and 12.5±8.4 ng μg−1 Chl a in Nostoc sp. ‘Macrozamia serpentina 73.1’ extracts. Further scans indicated the presence of the rare isoform [L-Har2] nodularin, which contains l-homoarginine instead of l-arginine. Nodularin was also present at 1.34±0.74 ng ml−1 (approximately 3 pmol per g plant ww) in the methanol root extracts of M. riedlei MZ65, while the presence of [L-Har2] nodularin in the roots of M. serpentina MZ73 was suggested by HPLC HESI-MS/MS analysis. The ndaA-B and ndaF genomic regions were sequenced to confirm the presence of the hybrid polyketide/non-ribosomal gene cluster. A seven amino-acid insertion into the NdaA-C1 domain of N. spumigena NSOR10 protein was observed in all endosymbiont-derived sequences, suggesting the transfer of the nda cluster from N. spumigena to terrestrial Nostoc species. This study demonstrates the synthesis of nodularin and [L-Har2] nodularin in a non-Nodularia species and the production of cyanobacterial hepatotoxin by a symbiont in planta.