Galois Theory: Main Theorem and Fundamental Correspondence | Graduate Course

Three classical geometric problems—trisecting an arbitrary angle, doubling the volume of a cube, and constructing a square with the same area as a circle—are impossible to solve using only a compass and unmarked straightedge because the required constructions involve numbers (cosine of 20°, cube root of 2, and square root of π) that cannot be expressed through algebraic operations with rational numbers where the degree of the minimal polynomial is not a power of two; this was proven using abstract algebra and the rational root theorem.

Four famous problems from ancient Greece remained unsolved for over 2,000 years: doubling the cube, trisecting an arbitrary angle, constructing a regular heptagon, and squaring the circle. These problems arose from Greek mathematicians' quest to determine what geometric constructions were possible using only ruler and compass. The resolution came in the 19th century when it was proved that all four are impossible. Ruler and compass constructions follow precise rules: starting with two points, you can draw lines through any two points (adding all intersections) or circles centered at one point passing through another (adding all new intersections). The four problems translate to algebraic questions: doubling the cube requires constructing cube root of 2; squaring the circle requires constructing sqrt(pi); trisecting angles relates to specific trigonometric values; constructing a regular heptagon relates to cos(360/7°).

The Three Classical Problems of Antiquity are: (1) Squaring the Circle: Construct a square with the same area as a given circle; (2) Doubling the Cube: Construct a cube with twice the volume of a given cube; (3) Trisecting an Angle: Divide an arbitrary angle into three equal parts. These were posed around 400 BC and remained unsolved for over 2000 years. Geometric construction with straightedge and compass corresponds to solving equations: the intersection of two lines gives a linear system solution, while the intersection of a line and circle or two circles gives a quadratic equation solution. Thus, constructible numbers are those obtainable by solving linear and quadratic equations. The Squaring the Circle problem was proven impossible in 1882 by Ferdinand von Lindemann, who showed π is transcendental. The Doubling the Cube problem was proven impossible in 1837 by Pierre Wantzel, who showed ∛2 cannot be constructed. The Trisecting an Angle problem was also proven impossible by Wantzel in 1837.

This video explains four famous geometric construction problems that ancient mathematicians could not solve using only a compass and straightedge: doubling the cube (constructing a cube with twice the volume), squaring the circle (constructing a square with the same area as a given circle), trisecting an arbitrary angle (dividing any angle into three equal parts), and constructing regular polygons with more than 6 sides (except for specific cases like the 17-gon). These problems were proven impossible in 1830 through the mathematical revolution initiated by Évariste Galois, which established that these constructions require solving algebraic equations that cannot be solved using only square roots, and that π and e are transcendental numbers that cannot be constructed with these tools.

A real number is constructible if and only if its field extension over the rational numbers has degree equal to a power of 2, which explains why the three classic Greek geometric problems (trisecting an arbitrary angle, squaring the circle, and doubling the cube) are impossible: trisecting a 60° angle would require constructing a 20° angle, whose cosine satisfies an irreducible cubic equation (degree 3, not a power of 2); squaring the circle would require constructing √π, which is transcendental (infinite degree); and doubling the cube would require constructing ∛2, which has degree 3 over the rationals.

This section presents general formulas for computing Galois group orders. For cyclotomic polynomials xⁿ - 1 over Q, the Galois group has order φ(n) (Euler's totient function), because the splitting field is Q(ζₙ) and the Galois group is isomorphic to (Z/nZ)×. For binomial polynomials xⁿ - 2 over Q, the Galois group has order nφ(n), combining the degrees of adjoining the root and the primitive nth root of unity. These formulas provide efficient methods for determining Galois group orders without full computation.

The Galois group of a polynomial g(x) over a field F is defined as the group of automorphisms of its splitting field K that fix F pointwise. This group acts faithfully on the set of roots of g(x), and there exists a one-to-one group homomorphism from the Galois group into the symmetric group S_n, where n is the degree of the polynomial. When g(x) is irreducible over F, this action is transitive, meaning the Galois group can map any root to any other root. The order of the Galois group is finite and divides n! (the factorial of the degree), and for irreducible polynomials, the degree n divides the order of the Galois group.

The Galois group of the polynomial x³-3x+3 over Q is S₃ (the symmetric group on 3 elements), determined by showing the polynomial has exactly one real root (using calculus to find critical points at x=±1 and evaluating the function at these points), which implies two complex roots, and applying Eisenstein's criterion with p=3 to prove irreducibility, which ensures the Galois group is transitive and contains a 3-cycle, and since it also contains complex conjugation (a 2-cycle), the group must be the full S₃.

This comprehensive methodology determines the Galois group of a quartic polynomial through two key tools: the cubic resolvent and discriminant analysis. The cubic resolvent R₃(x) = x³ - 8x - 16 is constructed from the original polynomial's coefficients. The discriminant formula Δ = -27a⁴ + 256b³ yields Δ = -4864 for f(x) = x⁴ - 4x + 2. When the discriminant is not a perfect square, the Galois group is determined by the resolvent's reducibility: if irreducible over Q, the group is S₄; if reducible, it is either D₄ or C₄. Reducing R₃(x) modulo 5 shows it has no roots in Z/5Z, proving irreducibility over Q. Combined with the non-square discriminant, this conclusively identifies the Galois group as S₄ without explicitly finding the roots.

For an irreducible cubic polynomial over a field K, the Galois group is either A₃ (cyclic of order 3) or S₃ (order 6). The group is A₃ if and only if the discriminant Δ = -4a³ - 27b² is a square in K. For an irreducible quartic polynomial, the Galois group is a transitive subgroup of S₄: S₄ (order 24), A₄ (order 12), D₄ (order 8), or V₄ (order 4). The group is determined by: (1) whether the discriminant is a square (distinguishing A₄ from S₄), and (2) whether the cubic resolvent R(x) = x³ - ax² - 4cx + (4ac - b²) is irreducible (distinguishing A₄ from D₄ and V₄).

Infinite Galois theory extends classical Galois theory to infinite-degree algebraic field extensions by defining a Galois extension as one that is algebraic, covered by finite Galois extensions, and has only the base field fixed by all automorphisms; unlike finite Galois theory, the correspondence between intermediate fields and subgroups of the Galois group fails in the infinite case because different subgroups can fix the same field, requiring the introduction of the Krull topology (a compact Hausdorff topology) on the Galois group to restore the correspondence, where closed subgroups correspond to intermediate fields.

This comprehensive lecture covers infinite Galois extensions, generalizing finite extensions to algebraic closures. Three equivalent conditions define infinite Galois extensions: separability and normality, being splitting fields of separable polynomials, and being fixed fields of group actions. The Krull topology defines the Galois group as an inverse limit of finite Galois groups, making it a profinite group. The Galois correspondence requires closed subgroups rather than all subgroups. Key examples include: the absolute Galois group of a finite field F_p is the profinite completion of Z (product of p-adic integers), and the cyclotomic extension of Q has Galois group equal to the product of units of p-adic integers. Z_p extensions are fields with Galois group Z_p, and the Kronecker-Weber theorem states Q has exactly one Z_p extension. Obstructions exist to extending Galois groups over Q, such as no cyclic extension of order 4 containing Q(i). The absolute Galois group of Q remains an open problem, with the Shafarevich conjecture suggesting all extension problems arise from the cyclotomic part.

The fundamental theorem of Galois theory extends to infinite Galois extensions. There is a bijection between intermediate fields of L/K and closed subgroups of Gal(L/K), reversing inclusion. For finite extensions, corresponding objects are open subgroups; for finite Galois extensions, they are open normal subgroups. This theorem relies crucially on the Krull topology on the Galois group.

The fundamental theorem of Galois theory extends to infinite algebraic extensions. The functions ψ (from intermediate subfields to subgroups of automorphisms) and φ (from subgroups to fixed fields) remain well-defined. For an intermediate subfield E, the fixed points of E-linear automorphisms of F̄ is precisely E itself. This is proven by contradiction: if an element α is fixed by all E-linear automorphisms but not in E, its minimal polynomial over E has degree ≥2, giving another root α' ≠ α. By the isomorphism extension theorem, there exists an E-isomorphism sending α to α', contradicting the assumption that α is fixed by all automorphisms. With respect to the Krull topology, the image of ψ consists precisely of closed subgroups, establishing a bijection between closed subgroups and intermediate subfields. There is also a bijection between normal intermediate subfields and closed normal subgroups, with the restriction map providing Gal(E/F) ≅ Gal(F̄/F)/ψ(E).

The Krull topology is a topology defined on the Galois group of an infinite Galois extension, where basic open sets are cosets of subgroups fixing finite intermediate fields. This topology resolves the breakdown of the Galois correspondence in infinite extensions by ensuring that intermediate fields correspond bijectively to closed subgroups of the Galois group. The topology is Hausdorff and makes the Galois group a topological group, with the closure of any subgroup being the subgroup of automorphisms that fix everything the original subgroup fixes.

This section covers the algebraic structure governing prime splitting: Frobenius automorphisms describing how primes behave in Galois extensions, decomposition groups capturing local behavior at each prime ideal, and the relationship between these objects and the global Galois group. The Frobenius element generates the decomposition group and determines how a prime splits in the extension.

Irreducible monic polynomials in Fq[T] correspond to prime ideals in the ring Fq[T]. The Galois group of any finite extension of a finite field is generated by the Frobenius automorphism (x → x^q). The orbits of the Frobenius automorphism on the algebraic closure correspond to irreducible polynomials. Given an orbit, the corresponding irreducible polynomial is the product (x-a) over all roots a in the orbit.

The Frobenius automorphism of a finite field can be lifted to characteristic 0 by considering an integral form of the field extension and reducing modulo primes; this technique demonstrates that -1 is a square modulo an odd prime p if and only if p ≡ 1 mod 4, as shown by examining the action of the Frobenius automorphism on the roots of the polynomial x² + 1.

In finite fields of characteristic p, the map x → x^p is a field automorphism called the Frobenius endomorphism, which preserves both addition and multiplication. The decomposition of prime ideals in cyclotomic fields can be expressed using ideal language. Any root of unity ζ ≠ 1 generates an ideal that contains p if and only if p divides the order of ζ. This relates to the decomposition of p in the cyclotomic field. Congruence modulo a prime ideal p is equivalent to congruence modulo p as an integer. The quotient ring Z[ζ_n]/p is a finite field isomorphic to F_p^f_p, where f_p is the order of p modulo n. The cyclotomic polynomial Φ_n(x) modulo a prime ideal p factors into irreducible polynomials of degree f_p, where f_p is the order of p modulo n.

This comprehensive section establishes the foundational framework of algebraic number theory. It begins with rings of integers in number fields and defines the class group as fractional ideals modulo principal ideals, with the class number measuring unique factorization failure. Gauss's conjecture (later proved) states only finitely many imaginary quadratic fields have class number 1. The section develops Galois-theoretic tools including the Frobenius automorphism and decomposition groups, explaining how primes split, remain inert, or ramify in extensions. These concepts form the essential vocabulary for understanding arithmetic properties of number fields and their extensions.