This book is the English translation (by D. Kramer) of the second edition of the Bewersdorff German book Algebra für Einsteiger: Von der Gleichungsauflösung zur Galois-Theorie (2004). The main aim of the book is to present Galois theory as the culmination of centuries-long investigations of solving algebraic equations by radicals. The book carefully describes results made in the first half of the 19th century involved in a long and complicated historical evolution, and the difficult transformation of classical methods used for algebraic equations solved by radicals into modern mathematical abstractions. Each chapter of the book begins with rhetorical questions or simple exercises illustrating important points of what lies ahead. Then the historical roots of investigations, their motivations, solutions, results and their applications are shown step by step.
Results are formulated first in an elementary way, then in their modern form (with the help of ideas and properties of groups and fields). The author describes solutions of cubic and biquadratic equations with their geometrical applications, including ancient problems, the birth of complex numbers, the discovery and proofs of the fundamental theorem of algebra, an attempt to solve equations of higher degrees and results of these attempts made by Paolo Ruffini, Etienne Bézout, Ehrenfried Walther, the Count of Tschirnhaus, Erland Samuel Bring, George Biích Jerrard, Niels Henrik Abel, Évariste Galois, Emil Artin, etc. The book can be recommended to undergraduate and graduate students.