Introduction to Abstract Algebra

Dr. Kuldeep Singh · Dr. Ankur Bala · Dr. Saurav Suman

2024년 10월 · Navi International Book Publication house

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Mathematicians who specialize in abstract algebra study algebraic structures like fields, rings, and groups. Abstract algebra investigates the fundamental ideas and patterns that underpin these procedures, as contrast to elementary algebra, which works with particular equations and operations on numbers. It is a fundamental topic with applications in computer science, cryptography, and physics. It also offers the theoretical basis for many other areas of mathematics. The idea of a group, which is a set with a single operation that meets axioms such as closure, associativity, the presence of an identity element, and the existence of inverses, is one of the fundamental ideas in abstract algebra. A common subject in the study of symmetry and transformations is groups. By adding new operations, including addition and multiplication, and examining their interactions, rings and fields expand on fundamental concepts. By studying abstract algebra, mathematicians may identify patterns and correlations that remain across many systems by moving from concrete numbers to more generalized things. This abstraction makes it possible to comprehend mathematical structures more deeply and inspires the creation of new ideas and instruments. As a field of study, abstract algebra serves as a doorway to more complicated mathematical analysis and as a potent language for characterizing intricate systems across a range of scientific fields. The importance of abstract algebra is not limited to mathematics alone; it also affects other practical disciplines. For example, in computer science, knowledge of abstract algebraic structures is essential to comprehending data structures, algorithms, and cryptographic systems. Group theory and field theory ideas play a major role in cryptography, which protects digital communications, in the creation and cracking of encryption systems. Similar to this, group theory's description of symmetry operations in physics aids in the explanation of key ideas in relativity and quantum mechanics. This field's intrinsic abstraction encourages other ways of thinking. It promotes the development of rigorous yet creative problem-solving abilities since it often calls for identifying patterns and generalizations that are not immediately apparent. This ability to think abstractly is useful not just in mathematics but also in other fields like economics, engineering, and biology that study complex systems. Because of its degree of abstraction and divergence from the arithmetic and algebraic intuition acquired in previous mathematics courses, abstract algebra may be difficult to understand in educational settings

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Dr. Kuldeep Singh serves as the Assistant Professor of Mathematics at Government College for Women, Ratia, Haryana. As an alumnus of Delhi University, he possesses a solid academic foundation in mathematics. With over nine years of dedicated teaching experience in higher mathematics, Dr. Singh has developed a reputation for effectively mentoring and guiding students at various stages of their academic careers. His expertise is further demonstrated by his success in clearing multiple national-level competitive exams, including securing AIR-43 in the CSIR-JRF, reflecting his deep knowledge and commitment to academic excellence. In addition to his teaching, Dr. Singh has made substantial contributions to the field of mathematics through his research, with several of his papers published in reputed international journals. He remains actively engaged with the wider academic community, having participated in more than 30 national and international seminars, workshops, and Faculty Development Programs. Dr. Singh’s blend of teaching excellence, research contributions, and academic leadership has earned him widespread respect within the field of mathematics.

Dr. Ankur Bala is a distinguished faculty member in the Department of Mathematics at Government College, Hansi (Haryana). With a solid academic foundation as an alumna of Delhi University, she has dedicated over nine years to teaching higher mathematics, earning a reputation for her expertise and commitment to student success. Her academic journey is marked by the successful clearance of multiple national-level examinations, including the highly competitive CSIR-NET, a testament to her deep understanding of the subject matter. Dr. Bala has further enriched the field of mathematics through her extensive research contributions, having published numerous papers in prestigious international journals, where her work has been recognized for its scholarly rigor and innovation. Beyond her research, Dr. Bala is actively involved in the academic community, participating in a wide range of national and international seminars and workshops.

Dr. Saurav Suman is a distinguished faculty member at DIPS Academy, Delhi, known for his expertise in higher mathematics. An alumnus of Delhi University, he has built a robust academic career over the past nine years, demonstrating a deep commitment to both teaching and research. His teaching methodology is highly regarded, and he has mentored countless students in mastering complex mathematical concepts. Dr. Suman’s academic excellence is further highlighted by his success in clearing several of India's most competitive national-level exams, including the CSIR-JRF, which is a testament to his deep understanding of the subject matter. Beyond teaching, he has made substantial contributions to the mathematical community through his research. His numerous papers, published in internationally recognized journals