In addition to elementary algebraic structures such as groups, rings and solids, Galois theory in particular is developed together with its applications to the cyclotomic fields, finite fields or the question of the resolution of polynomial equations.Special attention is paid to the natural development of the contents.

Motivation and prerequisites.- Field extensions and algebraic elements.- Groups.- Group quotients and normal divisors.- Rings and ideals.- Euclidean rings, principal ideal rings, Noetherian rings.- Factorial rings.- Quotient fields for integrality domains.- Irreducible polynomials in factorial rings.- Galois theory (I) - Theorem A and its variant A'.- Intermezzo - explicit example.- Normal fields extensions.- Separability.- Galois theory (II) - The main theorem.- Cyclotomic fields.- Finite fields.- More group theory - Group operations and Sylow.- Resolvability of polynomial equations.

This book contains the fundamental basics of algebra at university level.  In addition to elementary algebraic structures such as groups, rings and fields, the text in particular covers Galois theory together with its applications to cyclotomic fields, finite fields as well as solving polynomial equations. Special emphasis is placed on the natural development of the contents. Various supplementary explanations support this basic idea, point out connections and help to better comprehend the underlying concepts.  The book is particularly suited as a textbook for learning algebra in self-study or to accompany online lectures. The Author:  Prof. Dr. Marco Hien worked at the University of Regensburg after a postdoctoral year at the University of Chicago. Since 2010, he has been Professor of Algebra and Number Theory at the University of Augsburg with research interests in algebraic geometry and algebraic analysis. In 2020, he received the "Prize for Good Teaching" from the Bavarian Ministry of Science.The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content.

Develops the basic concepts and essential statements of abstract algebra up to the Galois theory step by step Contains a variety of motivational intermediate statements and encourages self-reflection Shows the effectiveness of the theory with many concrete examples