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 .

New series of 4, 4'-((2-(Aryl)-1H-benzo [d] imidazole-1, 3 (2H)-diyl) bis (methylene)) Diphenol (3a-g) was successfully synthesized from cyclization of the reduction product of bis Schiff bases (2) with aryl aldehydes bearing phenolic hydroxyl in the presence of acetic acid. The structure of these compounds was identified from FT-IR, 1H NMR, 13C NMR and EIMs. The Antioxidant capability was screened by DPPH and FRAP assays. Both assays showed antioxidant capability more than BHT as well. Compounds 3b and 3c showed antioxidant capacity slightly less than ascorbic acid. The docking study for theses compound was carried out as III DNA polymerase inhibitor. The results of docking demonstrated that the increase in hinderances around phenolic hydr

New series of 4,4'-((2-(Aryl)-1H-benzo[d]imidazole1,3(2H)-diyl)bis(methylene))Diphenol(3a-g) was successfully synthesized from cyclization of the reduction product of bis Schiff bases (2) with aryl aldehydes bearing phenolic hydroxyl in the presence of acetic acid. The structure of these compounds was identified from FT-IR, 1H NMR, 13C NMR and EIMs. The Antioxidant capability was screened by DPPH and FRAP assays. Both assays showed antioxidant capability more than BHT as well. Compounds 3b and 3c showed antioxidant capacity slightly less than ascorbic acid. The docking study for theses compound was carried out as III DNA polymerase inhibitor. The results of docking demonstrated that the increase in hinderances around phenolic hy

Compound 4-(((6-amino-7H-[1, 2, 4] triazolo [3, 4-b][1, 3, 4] thiadiazin-3-yl) methoxy) methyl)-2, 6-dimethoxyphenol (6) was synthesized by multi steps. The corresponding acetonitrile thioalkyl (7) was cyclized by refluxing with acetic acid to afford 4-(((6-amino-7H-[1, 2, 4] triazolo [3, 4-b][1, 3, 4] thiadiazin-3-yl) methoxy) methyl)-2, 6-dimethoxyphenol (8). Two new series of 4-(((6-(3-(4-aryl) thioureido)-7H-[1, 2, 4] triazolo [3, 4-b][1, 3, 4] thiadiazin-3-yl) methoxy) methyl)-2, 6-dimethoxyphenol (9a-c) and of 4-(((6-(substitutedbenzamido) 7H-[1, 2, 4] triazolo [3, 4-b][1, 3, 4] thiadiazin-3-yl) methoxy) methyl)-2, 6-dimethoxyphenol (10a-c) were synthesized as new derivatives for fused 1, 2, 4-trizaole-thiadiazine (8). The antioxidant

Four Co(II), (C1); Ni(II), (C2); Cu(II), (C3) and Zn(II), (C4) chelates have been synthesized with 1-(4-((2-amino- 5‑methoxy)diazenyl)phenyl)ethanone ligand (L). The produced compounds have been identified by using spectral studies, elemental analysis (C.H.N.O), conductivity and magnetic properties. The produced metal chelates were studied using molar ratio as well as sequences contrast types. Rate of concentration (1 ×10􀀀 4 - 3 ×10􀀀 4 Mol/L) sequence Beer’s law. Compound solutions have been noticed height molar absorptivity. The free of ligand and metal chelates had been applied as disperse dyes on cotton fabrics. Furthermore, the antibacterial activity of the produced compounds against various bacteria had been investigated. F

Compound 4-(((6-amino-7H-[1,2,4]triazolo[3,4-b][1,3,4]thiadiazin-3-yl)methoxy)methyl)- 2,6-dimethoxyphenol (6) was synthesized by multi steps. The corresponding acetonitrile thioalkyl (7) was cyclized by refluxing with acetic acid to afford 4-(((6-amino-7H-[1,2,4]triazolo[3,4- b][1,3,4]thiadiazin-3-yl)methoxy)methyl)-2,6-dimethoxyphenol (8). Two new series of 4-(((6-(3- (4-aryl)thioureido)-7H-[1,2,4]triazolo[3,4-b][1,3,4] thiadiazin-3-yl)methoxy)methyl)-2,6- dimethoxyphenol (9a-c) and of 4-(((6-(substitutedbenzamido)7H-[1,2,4]triazolo[3,4- b][1,3,4]thiadiazin-3-yl)methoxy)methyl)-2,6-dimethoxyphenol (10a-c) were synthesized as new derivatives for fused 1,2,4-trizaole-thiadiazine(8). The antioxidants of newly compounds were evaluated by DPPH

In this study new derivatives of O-[2-{''2-Substituted Aryl (''1,''3,''4 thiadiazolyl) ['3,'4-b]-'1,'2,'4- Triazolyl]-Ethyl]-p- chlorobenzald oxime (6-11)have been synthesized from the starting material p-chloro – E- benzaldoxime 1.Compound 2 was synthesized by the reaction of p-chloro – E- benzaldoxime with ethyl acrylate in basic medium. Refluxing compound 2 with hydrazine hydrate in ethanol absolute afforded 3. Derivative 4 was prepared by the reaction of 3 with carbon disulphide, treated of compound 4 with hydrazine hydrate gave 5. The derivatives (6-11) were prepared by the reaction of 5 with different substitutesof aromatic acids. The structures of these compounds were characterized from their melting points, infrared spectroscopy

Four Co(II), (C1); Ni(II), (C2); Cu(II), (C3) and Zn(II), (C4) chelates have been synthesized with 1-(4-((2-amino- 5‑methoxy)diazenyl)phenyl)ethanone ligand (L). The produced compounds have been identified by using spectral studies, elemental analysis (C.H.N.O), conductivity and magnetic properties. The produced metal chelates were studied using molar ratio as well as sequences contrast types. Rate of concentration (1 ×10􀀀 4 - 3 ×10􀀀 4 Mol/L) sequence Beer’s law. Compound solutions have been noticed height molar absorptivity. The free of ligand and metal chelates had been applied as disperse dyes on cotton fabrics. Furthermore, the antibacterial activity of the produced compounds against various bacteria had been investigated. F