Threelds, https://cp4space.hatsya.com/2022/05/25/threelds/

I have no idea whether it's useful for anything, but a threeld is a pair of fields where the multiplication operation on the inner one forms the multiplication on the outer one. The finite ones have inner order 3 and outer order 2, or inner order a Mersenne prime and outer order the adjacent power of two, but there also exist infinite ones with inner field of characteristic 0 and outer of characteristic 2.