The classic problems of antiquity:
- doubling cube
- trisecting angle
- squaring circle
...reduced to algebra in the 19th century;
first 2 are Galois theory (1st is very easy);
3rd required transcendence of \pi.

Rota said that all results known before 1800 now have a good proof
(I don't think he said "perfect"; presumably we can gain new
 perspective even on them, but *on their own terms*,
 they're solved)
...except for 1 piece of 1 proof.