Integers that are sums of two cubes in the cyclotomic Zₚ-extension

Let n be a cubefree natural number and p≥ 5 be a prime number. Assume that n is not expressible as a sum of the form x³+y³, where x,y∈ Q. In this note, we study the solutions (or lack thereof) to the equation n=x³+y³, where x and y belong to the cyclotomic Zₚ-extension of Q. As an application, consider the case when n is not a sum of rational cubes. Then, we prove that n cannot be a sum of two cubes in certain large families of prime cyclic extensions of Q.