In this paper, we characterize whether the pure sextic fields Q( 6 √ m) with square-free integers m ≡ ±1 (mod 9) have power integral bases or do not; if m ≡ 2, 3 (mod 4), then Q( 6 √ m) have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown.

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The outbreak of pandemic throughout the world has paralyzed all activities including the education system. As physical academic sessions have completely ceased so quickly it has been shifted to emergency remote teaching. The purpose of present research is to highlight the challenges and problems associated with taking online classes in developing a country like Pakistan during Covid-19. This study is mainly related to exploring the socioeconomic, health, and educational impacts of E-learning on students of Lahore, an important city of Pakistan. For this, qualitative data was collected by online google questionnaire as reported by 1,263 students of different institutes in Lahore. Pearson's chi-square tests were applied to test the associations and correlation among targeted variables by using SPSS. To better analyze the percentage share of different variables in terms of impacts of online classes spatial interpolation technique with pie charts was used through Arc GIS 10.6. The outcomes of the current research clearly show that there are many problems a student faces to attend an online class. The main problems are including improper study environment, inequality in the availability of gadgets, insufficient online study material, no access or slow internet speed, and load shedding. The above said challenges are negatively influencing the study of students in terms of minimum or no participation of students and proxy attendance in online class. Moreover, student's health is also reported as badly affected by attending online classes. Weak eyesight, overweight, sleep deprivation, and behavioral issues among students are found to be higher in this lockdown era. In the context of this research, it is recommended to reduce the timings of online class, training of students, easy availability of gadgets, and cost-effective student internet packages can be introduced to encourage the students and to make effective online classes.

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