In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

Our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

In order to achieve optimal plant growth and production, essential nutrients must be readily available in adequate quantities and in a balanced proportion to give a good yield, especially broccoli which has health benefits that may not be found in many other plants. For this purpose, this experiment was carried out during the seasons 2019/2020 in the botanical garden of the Department of Biology, College of Science for Women, University of Baghdad, to study the effects of nitrogen and sulphur and their interaction on eight parameters reflecting the overall traits of vegetative growth, yield, and chlorophyll content of broccoli Brassica oleracea L. (var. italic JASSMINE F1 Hybrid). A factorial design with three replicates was use

The current research aimed to conducting two experiments to study the effect of coating hatching eggs with nano-titanium dioxide (nano-TiO2) and nano-silica dioxide (nano-SiO2) particles and their mixture with carboxymethyl cellulose (CMC) on the characteristics of hatching percentage, embryo growth inside the egg. The study was conducted in the Department of Animal Production, College of Agriculture, Tikrit University for the period from 19/3/2023 to 17/9/2024. It aimed to evaluate the coating of hatching eggs with Nano-TiO2 and Nano-SiO2 particles and their mixture with carboxymethyl cellulose CMC on the qualities of hatching percentage, embryo growth inside the egg, as well as trying to obtain the best and longest storage method for fert

The research addresses a fundamental Islamic jurisprudential Purposeful issue, which is (Sharia), and to indicate the impact of this on Islamic jurisprudence, deriving rulings and extracting purposes, and to repel the illusion that this issue is only doctrinal, and clarifying the aspects of similarities and links between them by explaining the origin of deriving the purposes of Islamic Law (Sharia) through the meanings and wisdom learned from the texts and the explanation of the rulings. The rulings of Islamic Law (Sharia) have urged bringing benefits and repelling harms, and that the path to do so is reason and its production. I began the research by defining the purposes of Islamic Law (Sharia), then defining the rule of rational right

This experiment was carried out at a private field in the eastern Radwaniyah Baghdad for the fall season 2020/2021 and spring 2021 to study the effects of adding mineral fertilizers, spraying salicylic acid and amino acids on some growth traits and yield of industrial potato plants. 200 kg N h-1 , 100 kg P2O5 h-1, 100 kg K2O h-1 and F2 consist of 275 kg N h-1, 180 kg P2O5 h-1, 200 K2O h-1 and F3 consist of 350 kg N h-1, 360 kg P2O5 h-1, 300 K2O h-1 and salicylic acid in three concentrations of 0,50 and 100 mg L-1 ( S1, S2, S3) and amino acids in three concentrations of 0, 1.25 and 2.5 ml L-1 ( A1, A2 , A3) It was carried out as a factorial split plot experiment, where the fertilizer levels (F1, F2 and F3) are in the main plot and th

Background. Colorectal cancer, ranking second place in global cancer mortality, arises from diverse causes. There is growing recognition of the substantial involvement of the epigenetic modifications of histones at the DNA level in the occurrence of CRC. Aim. To assess the expression of p53, HDAC1, and HDAC3 proteins in a cohort of CRC patients and to analyze potential relationship between their expression and the stages of CRC progression. Materials and Methods. The retrospective investigation was carried out on 95 paraffin-embedded CRC tissue samples. The expression of p53, HDAC1, and HDAC3 was assessed immunohistochemically. Results. Notably, the expression of the p53 protein in CRC tissue samples exhibited a prominent correlatio

New vision of Mr. Abdul Mahdi Al-Muttafiqi in the light of unpublished documents