Foundations of Abstract Algebra and Galois Theory: A Comprehensive Treatment of Group Structure, Field Extensions, Solvability by Radicals, and the Fundamental Theorem—From Historical Development to Modern Applications

This comprehensive treatise provides a rigorous development of abstract algebra from first principles, focusing on the revolutionary contributions of Évariste Galois to the theory of field extensions and polynomial solvability. We begin by establishing the foundational structures of group theory—including groups, subgroups, normal subgroups, quotient groups, and group homomorphisms—before constructing the algebraic framework of ring and field theory essential for understanding algebraic extensions. The central achievement of this work is a complete proof of the Fundamental Theorem of Galois Theory, which establishes a bijective correspondence between the intermediate fields of a Galois extension L/KL/KL/K and the subgroups of the Galois group Gal(L/K)\mathrm{Gal}(L/K)Gal(L/K), with subgroup normality corresponding precisely to normality of field extensions. We present Galois’ revolutionary solvability criterion, proving that a polynomial f(x)∈Q[x]f(x) \in \mathbb{Q}[x]f(x)∈Q[x] is solvable by radicals if and only if its Galois group is a solvable group—that is, one possessing a composition series whose factor groups are abelian. This result resolves the centuries-old problem of determining which polynomial equations can be expressed through nnnth roots and arithmetic operations. Applications include rigorous demonstrations of the Abel–Ruffini theorem (showing the general quintic is unsolvable by radicals), as well as proofs of the impossibility of certain classical geometric constructions (angle trisection, cube duplication, and circle squaring), and the theory of cyclotomic extensions. The historical development is traced from Cardano’s cubic formula (1545) through Lagrange’s resolvent method (1770) and Abel’s proof of quintic insolvability (1824), culminating in Galois’ posthumously published memoir (1846), which unified prior results through the lens of group-theoretic symmetry. Modern reformulations by Dedekind, Artin, and contemporary algebraists are examined in detail. We provide full proofs of the Tower Law for field extensions, separability and normality criteria, the Primitive Element Theorem, the fundamental properties of splitting fields, and the characterization of solvable groups via composition series. Extensive worked examples illustrate the theory, including cyclotomic polynomials, radical extensions, constructible numbers, and explicit Galois group calculations for polynomials of degrees 2, 3, 4, and 5. This treatise serves both as an introduction for graduate students entering the study of modern algebra and as a comprehensive reference for researchers requiring a complete, rigorously developed treatment of the machinery of Galois theory.