If the roots are cut and the cactus removed but some roots remain below ground will the cactus regrow?

Yes, cactus plants have roots.

Specifically, a cactus plant needs roots to take in moisture and dissolved nutrients from the soil. It is upon this underground supply of moisture and dissolved nutrients that the entire plant depends for survival and growth. Additionally, roots keep the plant anchored in the ground or - in the case of epiphytic cactus plants - to the branches of trees.