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 .

Background: Enforcement of sustainable and green chemistry protocols has seen colossal surge in recent times, the development of an effective, eco-friendly, simple and novel methodologies towards the synthesis of valuable synthetic scaffolds and drug intermediates. Recent advances in technology have now a more efficient means of heating reactions that made microwave energy. Efforts to synthesize novel heterocyclic molecules of biological importance are in continuation. Microwave irradiation is well known to promote the synthesis of a variety of organic and inorganic compounds. The aim of current study was to conceivea mild base mediated preparation of novel Schiff base of 2-Acetylpheno with trimethoprim drug (H2TPBD) and its complexes w

The Ligand 6,6--(1,2-benzenediazo) bis (3-aminobenzoicacid) derived from o-phenylenediamine and 3-aminobenzoicacid was synthesized. The prepared ligand was identified by Microelemental Analysis, 1HNMR, FT-IR and UV-Vis spectroscopic techniques. Treatment of the ligand with the following metal ions (CoII, NiII, CuII and ZnII ) in aqueous ethanol with a 1:1 M:L ratio and at optimum pH. Characterization of these compounds has been done on the basis of elemental analysis, electronic data, FT-IR and UV-Vis, as well as magnetic susceptibility and conductivity measurements. The nature of the complexes formed were studied following the mole ratio and continuous variation methods, Beer's law obeyed over a concentration range (1×10-4 - 3×10-4 M). H

It  can be said that the Security of water  in  Basra from the visual task vital strategic issues of concern to the attention of researchers in various attributions and those interested in water, environmental, economic, social, cultural and political affairs ... etc. This view of the great importance of the issue of water in the occupied Basra, which is characterized by parochialism and scarcity, When looking at the sources of our daily lives and in our reality today. We find that millions of people living on the two main exporters Tabaaan oil and water. And depleted oil wealth However Manfred him the most attention because we entered it surpasses all other sources of income, but is not it a littl

Industrial characteristics calculations concentrated on the physical properties for break down voltage in sf6, cf4 gases and their mixture with different concentrations are presented in our work. Calculations are achieved by using an improved modern code simulated on windows technique. Our results give rise to a compatible agreement with the other experimental published data.