A radical integer is a number obtained by closing the integers under addition, multiplication, subtraction, and root extraction. An example of such a number is 7^(1/3) + sqrt(-2) - sqrt(3 + (1 + sqrt(2))^(1/4)). The radical integers are a subring of the algebraic integers. There exist cubic algebraic integers which are not radical integers, namely those which can't be expressed in terms of radicals. R. Schroeppel proved that these are the only ones; i.e., if an algebraic integer can be expressed in terms of radicals, then it can be done so without using division.

algebraic integer | algebraic number | Euclidean number