Open questions and modern directions -- from inverse Galois to algebraic number theory
The classical theory raises as many questions as it answers. The inverse Galois problem asks: given a finite group G, is there a polynomial over Q whose Galois group is G? This question remains open in full generality. Meanwhile, the absolute Galois group Gal(Q-bar/Q) -- the symmetry group of all algebraic numbers -- is one of the most mysterious and important objects in modern mathematics.
In this lesson, explore solved cases of the inverse problem, visualize p-adic numbers as a gateway to modern number theory, and glimpse the infinite structure of the absolute Galois group.
For many finite groups, explicit polynomials over Q are known whose Galois groups realize that group. Browse a catalog of small groups -- cyclic, dihedral, symmetric, alternating -- and see their realizing polynomials and subgroup lattices.
Every finite group below is realized as a Galois group over Q. Click a group to inspect its subgroup lattice, roots, and automorphisms.
Key insight: Every symmetric group S_n appears as Gal(f/Q) for some polynomial f (in fact, the "generic" degree-n polynomial has Galois group S_n). Every abelian group appears by the Kronecker-Weber theorem. But for many non-abelian groups -- including some simple groups -- finding explicit realizing polynomials is an active area of research.
The p-adic numbers Q_p complete the rationals with respect to a metric where "close" means "divisible by a high power of p." The p-adic integers Z_p form a fractal-like nested structure: p residue classes, each subdivided into p sub-classes, infinitely deep. This perspective is central to modern Galois theory.
Fractal structure of the p-adic integers. Each residue class mod p^k nests inside its parent class mod p^(k-1).
Two p-adic integers are "close" when they agree modulo p^k for large k. Visually, this means they lie in the same nested sub-disc at deeper and deeper levels. The p-adic absolute value |x|_p is small precisely when p divides x to a high power.
Key insight: In the p-adic metric, 1 and 1 + p^10 are very close (they agree mod p^10), while 1 and 2 might be far apart. This "ultrametric" topology gives Z_p a totally disconnected, self-similar structure. Local-global principles connect p-adic solutions to rational solutions, making p-adic fields indispensable in modern number theory.
The absolute Galois group Gal(Q-bar/Q) is the projective limit of all finite Galois groups over Q. It is a profinite group of uncountable cardinality, yet it encodes all the arithmetic of the rationals. Click to zoom through successive approximations and watch the structure grow in complexity.
Gal(Q-bar/Q) is a profinite group -- the projective (inverse) limit of all finite Galois groups over Q. Add levels to watch the approximating group grow.
Key insight: Every finite quotient of Gal(Q-bar/Q) is the Galois group of some finite extension of Q. The full group is obtained as the limit of all these finite groups, connected by surjective maps. Understanding Gal(Q-bar/Q) is one of the deepest goals of algebraic number theory -- Grothendieck called it "the most remarkable mathematical object known."