Open questions and modern directions -- from inverse Galois to algebraic number theory
Which groups can be the Galois group of some polynomial? Nobody knows in general. Here are three doors into the research that grew out of Galois's ideas.
Click a group in the catalog to see a polynomial that realizes it and its subgroup lattice.
Every finite group below is realized as a Galois group over Q. Click a group to inspect its subgroup lattice, roots, and automorphisms.
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.
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.
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.
Pick a prime and a depth, then click a disc to zoom into it. Notice the same pattern of p smaller discs repeats at every level.
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.
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.
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.
Press Zoom In to add finite Galois groups one level at a time, and Zoom Out to remove them. Watch the groups grow quickly in size and 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.
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.
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."
Active research -- Galois theory connects to the Langlands program, etale cohomology, and arithmetic geometry
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