Galois theory is a second course, not a first one. The central theorem is a correspondence between the subgroups of a finite group and the intermediate fields of an extension, so you need both halves before the statement means anything: groups with normal subgroups, quotients and solvability, and field extensions understood as vector spaces with a dimension. People who try to read a Galois theory text cold almost always stall in the first chapter, and it is a prerequisite problem rather than a difficulty problem.
The other obstacle is bibliographic. Four books on this path carry essentially the same title, and they are wildly different in level: Ian Stewart's Galois theory, Emil Artin's Galois theory, David Cox's Galois Theory and Harold Edwards's Galois theory. Author names do all the work below.
Why anyone cared
Mario Livio's The Equation That Couldn't Be Solved is the popular history of symmetry and the quintic, and it is the cheapest way to understand what problem Galois was solving. Peter Pesic's Abel's Proof is shorter and closer to the mathematics, covering Abel's demonstration that the general quintic has no solution in radicals, with a translation of the paper appended.
Leopold Infeld's Whom the gods love is a novelised 1948 biography of Galois, and it deserves a caveat. Galois did die in a duel at twenty in 1832, but the familiar story — a genius scribbling the whole theory on the night before his death — is substantially legend. He had submitted memoirs to the Academy years earlier, and what he wrote that night were annotations and a letter to a friend. The historian Tony Rothman documented in the 1980s how much of the standard account was embroidered by later biographers. Read Infeld as an artefact of that romantic tradition rather than as history.
The algebra you need first
Charles Pinter's A Book of Abstract Algebra is the gentlest genuine introduction, cheap and built out of short chapters with worked exercises. M. A. Armstrong's Groups and symmetry comes at group theory through geometry and is unusually good at making solvability feel like something rather than a definition. Nathan Carter's Visual group theory is the intuition builder — Cayley diagrams, lots of pictures, and the clearest treatment of what a normal subgroup is doing. Joseph Gallian's Contemporary Abstract Algebra is the standard American course text and covers rings and fields as well as groups, which makes it the most efficient single prerequisite book if you only want one.
The floor for what follows: groups, normal subgroups, quotient groups, solvable groups, polynomial rings, irreducibility criteria, and vector space dimension.
Galois theory itself
Jorg Bewersdorff's Galois theory for beginners demands the least background and works through the classical problems concretely; it is the right book if the prerequisite list above still looks intimidating. Ian Stewart's Galois theory is the standard accessible course book, chatty, historically grounded, and revised repeatedly over decades — for most readers this is the main text.
Emil Artin's Galois theory is the famous Notre Dame lectures: about eighty pages, built on linear algebra, and beautiful. It is also terse to the point of austerity and assumes a reader who already knows where the argument is going, so it works far better as a second pass than a first. David Cox's Galois Theory is the opposite — long, generous with examples and exercises, carrying the history alongside the mathematics, and going on to solvability by radicals and straightedge-and-compass constructions in detail. If you want one comprehensive book, this is it.
Beyond the first course
Patrick Morandi's Field and Galois theory is a graduate text covering infinite extensions, transcendental extensions and inseparability in positive characteristic. Jean-Pierre Tignol's Galois' theory of algebraic equations traces the whole development from Cardano through Lagrange to Galois, and is the best book here for understanding why the theory took the form it did. Harold Edwards's Galois theory follows Galois's own memoir closely and takes a constructive approach, including a translation of the original text — an unusual and rewarding way to finish.
Follow the full ordered path here: Best Books to Learn Galois Theory, in Order.
