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 accounting system of government is considered an important tool to follow up the financial transactions that reflect the activities of governmental units and by which the useful information for estimating governmental annual revenues and expenditures are provided through the state public budget because it is an information system that provides detailed past performance, as well as measures the efficiency of the governmental agencies performance in implementing the budget, and the of success governmental units is measured through the type of services and programs offered, their size and the possibility of achieving the objectives assigned to them. The medical evacuation program is one of the medical and curative health services provid

Background: The association between facial types and dental arches forms has considerable implications in orthodontic diagnosis and treatment planning. The aim was to establish the maxillary and mandibular dental arches width and length in skeletal and dental class II division 1 and class III malocclusion groups, find out the most frequent dental arch form and facial type and the association between them and to check the gender differences. Materials and Methods: Frontal and lateral facial photographs and maxillary and mandibular occlussal photographs for 90 iraqi subjects with age 18-25 years old (45 males and 45 females) divided equally into three groups, the 1st group with class II division 1malocclusion (overjet more than 3mm but less t

The objective of the research was to evaluate consumer purchasing behavior through the Internet, such as consumer behavior, reasons for buying online, purchasing advantages over the Internet, personal variables (gender, age, marital status, education level, income, and income and job type). The questionnaire was adopted as a main tool in the survey of the views of a sample of consumers in Baghdad governorate (100) people and analyzed their answers using the statistical program SPSS in calculating the mean and standard deviation Centigrade, correlation coefficient (R) and test ( ). The main findings of the research were:

1. There is a positive and positive relationship between consumer purchasing behavior via the Internet and

When the number of confirmed coronavirus disease cases rose in Iraq in the middle of February 2021, the Iraqi government performed a closure approach to constrain mobility and factory operations and enforce social distancing. In this research, the concentrations of air components (PM_2.5, PM₁₀, nitrogen dioxide (NO₂) and ozone (O₃)), which represent herein the degree of air quality index, were recorded, drawn and evaluated over central (Baghdad, the capital), northern (Kirkuk Province) and southern (Basra Province) Iraq before and during the closure. The experimental duration of this research was 6 months (from 1 January 2021 to 30 June 2021), which

A cut-off low is a closed low with a low value of geopotential height at the upper atmospheric levels that has been fully detached (cut-off) from the westerly flow and move independently. A cut-off low causes extreme rainfall events in the mid-latitudes regions. The main aim of this paper is to investigate the cut-off low at 500 hPa over Iraq from a synoptic point of view and the behavior of geopotential height at 500 hPa. To examine the association of the cut-off low at 500 hPa with rainfall events across Iraq, two case studies of heavy rainfall events from different times were conducted. The results showed that the cut-off low at 500 hPa with a low value of geopotential height will strengthen the low-pressure system at the surface, lea