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 .

The rapid growth of cities and their inflation is a big problem, especially in the last years. this growth is accompanied by such problems like population growth, housing need, low level of services, random expansion, traffic congestion as well as pollution of the environment, which leads to a decline in the quality of life in Baghdad, the population are concentration in Baghdad therefore that leads to imbalance of development among cities and productive concentration for service projects in a mega cities, causing migration from other provinces In search of a better life. The main objective of the new cities is to relieve pressure on major cities and guide the growth of cities. Basmaya city it’s a new city project adopted f

The research aims to delve into the nature of international financial reporting standards and the unified accounting system adopted in the Iraqi environment and financial ratios in the theoretical side. In the practical aspect, the results of some financial ratios of Basrah Gas Company were compared with the adoption of the financial statements prepared on the basis of the consolidated accounting system with the prepared financial statements Based on the International Financial Reporting Standards (IFRS). The research has reached a number of conclusions, including a difference in the accounting rules and practices between IFRS and the consolidated accounting system in force in Iraq. The adoption of IFRS has led to a decrease in profitabi

This research is determined by the study of  the " cognitive references of the directorial imagination and modeling of  the theatrical actor performance ." it has described an Iraqi theatrical model, The research began with the great importance of the director's imagination as the basic premise for crystallizing the  director's vision according to its cognitive references in creating a solid performance model based on the aesthetic, intellectual and technical bases, It is also contributes to the formation of the theatrical show as a technical framework that presents the show in one unified fabric.

The research sought to reach through the problem of research, which is in the question of: What is the modeling of the

       The current study aimed at identifying the impact of each of the full and part time summer enrichment programs on the performance of gifted students. Moreover, it aimed to study the difference between the full and part time programs on the performance of gifted students. The study sample consisted of (115) students from the full time programs and (137) students from the part time programs, they have been randomly selected from the gifted students participating in the full and part time summer enrichment programs. The researcher used the scale of student performance. The results indicated that there were statistically significant differences between the averages of the pre and post applications of the

The current research aims to know the extent of the impact of performance evaluation in its dimensions as an explanatory variable in the behavioural and attitudinal work outputs with its dimensions as a response variable in order to reach appropriate solutions through which the University of Fallujah seeks to achieve its goal in the process of diagnosing the axes of strength and to benefit from them in the process of strengthening the status and sobriety of the academic position of the professor and the researcher relied on The descriptive and analytical approach in carrying out this study, and data was collected from university professors, including leaders, heads of departments and divisions, who numbered (97) teachers. And fie

In this paper, the dynamics of scavenger species predation of both susceptible and infected prey at different rates with prey refuge is mathematically proposed and studied. It is supposed that the disease was spread by direct contact between susceptible prey with infected prey described by Holling type-II infection function. The existence, uniqueness, and boundedness of the solution are investigated. The stability constraints of all equilibrium points are determined. In addition to establishing some sufficient conditions for global stability of them by using suitable Lyapunov functions. Finally, these theoretical results are shown and verified with numerical simulations.

The study aims to identify the mechanical and electrical activities of the heart according to the energy systems of advanced players and to detect the differences between the energy systems in terms of the mechanical and electrical activities of the heart for advanced players. It was clear from the results of the significance of the differences between the three groups according to the energy systems of the advanced players in all research variables that (the non-oxygenic system "Lactic"), which represents the advanced players in the arches (800 m, 1500 m) was the first in most tests of mechanical and electrical activities of the heart, which is (Margaria-Kalamen, Wingate, systolic muscle strength of the heart FC, Stroke Volume SV