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 present study attempts to examine verbal/nonverbal impoliteness in the classroom interaction and outside it in one of the movies, namely, The Marva Collins. Impoliteness, which is significantly studied within pragmatics, is a negative attitude towards particular behaviors. It always presumes to have emotional concerns for at least one participant who has caused it. This study is an attempt to examine verbal/nonverbal impoliteness in The Marva Collins movie. The study aims to investigate the different types of impoliteness strategies used in the four selected scenes and find out whether the speaker’s status has anything to do with the types of impoliteness. Besides, the functions performed are also examined by following a qual

The report includes a group of symbols that are employed within a framework that gives a language of greater impact. This research discusses the problem of the semiotic employment of religious symbols in press reports published in the electronic press across two levels: Reading to perceive the visual message in its abstract form, and the second for re-understanding and interpretation, as this level gives semantics to reveal the implicit level of media messages through a set of semiotic criteria on which it was based to cut texts to reach the process of understanding and interpretation.

The report includes a group of symbols that are employed within a framework that gives a language of greater impact. This research discusses the p

Jurisprudential disagreements are a significant source of a vast jurisprudential wealth. Among the manifestations of these disagreements are scholarly critiques raised by some scholars against others, claiming their divergence from the legal evidence or established principles for deducing legal rulings. One such critique is presented by the narrator scholar Abu Bakr Ibn Abi Shaybah, may Allah have mercy on him, directed towards Abu Hanifah al-Nu‘man, may Allah have mercy on him. In his compilation, Ibn Abi Shaybah cited numerous hadiths and traditions that he considered conflicting with the judgments as perceived by Imam Abu Hanifah, may Allah have mercy on him. These issues cover various branches of jurisprudence, including worship, sale

To evaluate the toxicity of benzalkonium chloride in aquaculture, the hemato-serological indices of Nile tilapia Oreochromis niloticus are used as biomarkers. Following exposure to three concentrations of benzalkonium chloride BAC 0.1, 0.25, 0.50, and 1 mg/l (BAC1,2,3 and 4) in aquaria for two durations 21 and 42 days, the microbiological assay in fish aquaria, in addition to blood parameters were assessed. Except for the mean difference between BAC2 and BAC3 (P > 0.05) at 42 days, the mean values of the bacterial counts revealed a significant difference between all compared groups (0.05 ≥ P ≤ 0.01). Following exposure to the lower concentrations of BAC (1, 2 and 3), the main blood parameters of Oreochromis niloticus namely red bl