This book covers topics in the history of modern algebra. More precisely, it looks at some topics in algebra and number theory and follows them from their modest presence in mathematics in the seventeenth and eighteenth centuries into the nineteenth century and sees how they were gradually transformed into what we call modern algebra. Accordingly, it looks at some of the great success stories in mathematics: Galois theory—the theory of when polynomial equations have algebraic solutions—and algebraic number theory. So it confronts a question many students ask themselves: how is it that university-level algebra is so very different from school-level algebra?