Yesterday Jack Kennedy told me about a graph he’d made as part of a project he’s working on and I asked if I could post it here.
The Galois group of a quadratic polynomial x2 + bx + c is either A2 or S2. If b2 – 4c is a perfect square, the polynomial has rational roots and the Galois group is the trivial group A2. Otherwise there are distinct irrational roots and the Galois group is the two-element group S2.
As b and c range over integers, color a pixel yellow if the group is A2 and black otherwise. This produces the image below.
Note that what appear to be the crossed lines y = ±x intersecting at 0 are actually the lines y = ±(x+1) intersecting at (-1,0).