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This book contains the basics of abstract algebra. 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. Numerous intermediate explanations support this basic idea, show connections and help to better penetrate the underlying concepts. The book is therefore particularly suitable for learning algebra in self-study or accompanying online lectures. Preface Motivation and Prerequisites Field Extensions and Algebraic Elements Groups Group Quotients and Normal Subgroups Rings and Ideals Euclidean Rings, Principal Ideal Rings, Noetherian Rings Unique Factorization Domains Quotient Fields for Domains Irreducible Polynomials in UFDs Galois Theory (I)—Theorem A and Its Variant A’ Intermezzo: An Explicit Example X5 – 777X + 7 Normal Field Extensions Separability Galois Theory (II)—The Fundamental Theorem Cyclotomic Fields Finite Fields More Group Theory—Group Actions and Sylow's Theorems Solvability of Polynomial Equations Proof of the Existence of an Algebraic Closure Tricks and Methods to Classify Groups of a Given Order