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 research problem is that the traditional methods of internal auditing are somewhat heavy with long and rigid procedures for the members of the audit process team, especially in light of the current developments that are reflected in the business environment and internal audit reports, so it is necessary to reconsider the traditional internal audit work method and assess the extent of its development by agile methods to reduce the time of the audit process on the activities and elements that add value and direct the effort and time to the activities and elements that add value to the work of the economic unit and the report of the internal auditor.

The research aims to study the possibility of applying agile internal auditing

     This study deals with the petrology of the lower succession of the Injana Formation in the Shorr Shareen area, Wasit Governorate, Eastern Iraq. The study revealed that the sandstone is litharenite consists of 45.56% rock fragments, 22.13% quartz and 8.5% feldspars. The matrix is about 8.39%, consisting of silt and clay particles. The cement is variable (carbonates 8.42%, evaporites 1.78% and iron oxides 0.96%). The grain assemblage infers that the source of the rock fragments is nearby. The petrographic analyses indicate that the studied Injana sandstones are immature mineralogically because of their content of unstable constituents, such as lithic fragments and feldspars. In addition, the presence of such fresh feldspars indica

Iraq has had more than 10000 km2 of geographical low land areas called marshes.
Enriched with great diversity of natural vegetation and wild life. With increasing
climatic changes and passive man interference phenomena, vast areas of these
marshes have deteriorated through drying out processes at an alarming rate.
According to recent survey achieved by several Iraqi ministries marshes areas have
decreased to about quarter of theS original area. The statistical data and geospatial
information are weak. We monitored, assessed the environmental processes and
detect changes using digitally processed landsat MSS (Multispectral Scanner) and
Spot (System Pour Observation) satellite images that transform haur Ibn Najm

Imposed on foreign oil companies from important sources in the financing of the general budget in most countries of the world income tax is considered as well as be used to achieve political, economic and social goals, and has developed the concept of the tax until it became play an important role in influencing the economic conditions of a country, and the aim of this research is to statement imposed on foreign oil companies operating in Iraq in the financing of the state budget income tax contribution, as well as clarify the contracts type contracts with these companies, which is in favor of Iraq, together with the Income Tax Law No. (19) for the year / 2010, and instructed No. (5) for the year / 2011, which organized the tax process s

The aim of this study is to investigate the sedimentation environments and diagenetic processes of the Ibrahim Formation (Oligocene-early Miocene) in Zurbatiya, eastern Iraq. The Ibrahim Formation is comprised mostly of clayey micrite and skeletal grains composed of planktonic foraminifera, calcispheres, radiolaria, and benthic foraminifera. Glauconite and pyrite were documented in some restricted zones of this formation; they reflect quiet and reducing conditions. Radiolaria were identified in Late-Oligocene which was not known previously at this age regionally in carbonate formations of the Arabian Plate (AP). Mudstone, wackestone, and planktonic foraminiferal wackepackstone are the main microfacies that are affected by dissolutio

The Sarmord Formation is one of the overlooked formations that require more investigations than that achieved before. There are many problems in its definition, age, stratigraphy and aerial distribution. Previously in most parts of the Sulaymaniyah, it is not mapped and its outcrops combined with Balambo Formation while in the present study it is differentiated, mapped in the field on the Qaywan anticline and its problems discussed in details with a suggestion of solutions. For laboratory studies, 31 samples are taken on the Qywan Mountain (anticline) at 23 km north of the Sulaimani city for nannofossils biozonation. This study identified 19 species of the recorded calcareous nannofossils, the studied section reveals three biozones

The Paleocene-Early Eocene sequence is represented by Aliji and Umm Er Radhuma formations, while the Middle-Late Eocene sequence is represented by Jaddala and Dammam formations. The Rus Formation has been described and its basin was analyzed separately because it was deposited during the regression period (Middle Eocene), which is a transitional period between these two cycles.  

This study includes analysis of the geohistory of this succession,  interpretation of the changes of the accumulation, and calculation of subsidence rates. The results were compared with the space available to explain the basin development. The study site included the boreholes of Garraf-84 and 92, Halfaya-1, Nasirya-13 and 40, and Noor-5 at th