Elementary Galois Theory

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This textbook addresses classical mathematical questions such as the impossibility of squaring the circle and the division of angles with compass and ruler, as well as the absence of general solution formulas for polynomial equations of degree 5 and higher. It presents these topics through the lens of Galois theory, offering both a classical introduction and practical applications. Adopting a constructive mathematician's perspective, it ensures that the existence of mathematical objects is accompanied by algorithmic constructions. Consequently, some statements are presented with more caution than usual, and proofs are elaborated for clarity and comprehension. Abstract theories and definitions emerge from concrete problems, enhancing understanding. The material is designed for a one-semester algebra course, making it suitable for first-year Bachelor's students and teachers alike. Key statements are summarized within the text to encourage reflection and targeted review. Each chapter concludes with a concise summary of essential arguments and a variety of exercises that increase in difficulty. The translation utilized artificial intelligence, followed by a human revision focused on content accuracy.