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 .

Atheists have spread in the modern era, so that atheism has become a bad phenomenon in the world in general and in Islamic societies in particular, so the research aims to study the individual and social effects left by atheism on the atheists themselves, and the research included multiple axes: atheism linguistically and idiomatically, atheism in the Qur’an Noble and Modern (and Contemporary) Atheism Statistics: and the reasons for atheism: Studying the phenomenon of atheism in Iraq as a model, then studying the effects of atheism: on the individual first, then atheism and its impact on society, then the conclusion, recommendations, sources and references

يشهد المسرح الدولي قضايا مستجدة ومستحدثة مثل تعدد الفاعلين من الدول ومن غير الدول وتزايد مساحات نفوذهم وتأثيرهم ،ومن جانب اخر أثرت جائحة كورونا على البيئة والمناخ بشكل ملموس على جوانب عدة. فضلا عن ذلك تأثرت جميع المجتمعات البشرية بالإرهاب والهجمات الالكترونية وغيرها ومن بينها دول والمجتمع الدولي بإفراده مؤسسات وكيان اقتصادي وسياسي وقانوني وأمني وامتداد اثاره واسبابه الكثيرة والمتداخلة وما يثيره من خلق حا

A study was conducted to evaluate the antibacterial effect of Phyllanthus emblica extract (ethanol:methanol, 1:1) against Pseudomonas aeruginosa, Staphylococcus aureus and Escherichia coli at different concentrations, i.e. 0.625, 1.25, 2.50, 5.0, 10.0 and 20.0 mg/ml. The antibacterial activity was determined by the agar well diffusion method to investigate the minimum inhibitory concentration (MIC) and minimum bactericidal concentration (MBC). The alcoholic extract of Phyllanthus emblica had the highest antibacterial activity at 20 mg/ml and 5 mg/ml except for Pseudomonas aeruginosa where the value of inhibition was between 20 and 10 mg/ml. The MIC concentrations were mostly very high and ranged from 5 to 1.25 mg/ml, while the MBC range fro

The present study aims at scrutinizing the impoliteness types, causes, and purposes utilized by Iraqi English language learners when refusing marriage proposals. Thus, it attempts to answer the questions: (1) what are the impoliteness formulas used by the Iraqi learners of English in refusing marriage proposals?, and (2) What are their impoliteness triggers/causes and the purposes? The study is significant in bridging the gap that few linguistic types of research concentrate on studying intentionality and emotions allied with impoliteness. Data were collected from 35 Iraqi learners of English responding to 6 situations of marriage. The data were analyzed using Culpeper’s (2011) formulas of impoliteness and Bousfield’s (2007) impolite

     The research aimed to identify the extent to which the media offices in the ministries depend on the website of the Iraqi Council of Ministers as a source of information. Such goal includes a set of sub-goals: Knowing the reasons for media offices' reliance on the information provided by the Cabinet's website; and revealing the benefit derived from relying on the Cabinet's website.

The researcher used the survey method to screen and comprehend the extent to which the media offices in the ministries depend on the website of the Iraqi Council of Ministers as a source of information.

The mechanism of comprehensive account of the limits of research wa

Endophytic fungi live inside plants or any part of them without creating any visible pathogenic signs. Endophytic fungi are found within medicinal plants and have shown strong biologic activity, such as anticancer and antioxidant activities, as well as producing extracellular enzymes. In this study, different fungal strains were isolated from the leaves of the medicinal plant Ziziphus spina, including Aspergillus flavus, Aspergillus fumigatus, Aspergillus niger, Cladosporium sp., Rhizopus sp., and Mucor sp. Extracellular enzymes have been quantified using agar plate-based methods in which fungi were grown in specified growth media to detect the enzymes produced. The results showed that A. niger has the highest ability to produce amy

            Liquid – liquid equilibria data were measured at 293.15 K for the pseudo ternary system (sulfolane + alkanol) + octane + toluene. It is observed that the selectivity of pure sulfolane increases with cosolvent methanol but decreases with increasing the chain length of hydrocarbon in 1-alkanol. The nonrandom two liquid (NRTL) model, UNIQUAC model and UNIFAC model were used to correlate the experimental data and to predict the phase composition of the systems studied. The calculation based on NRTL model gave a good representation of the experimental tie-line data for all systems studied. The agreement between the correlated and the experimental results was very good