Question

Prove that cube root of p is irrational

Answer 1

 Let p be prime. Then assume by contradiction that cube root of p is rational. Then cube root of p = m/n for integers m and n with n =/= 0. Assume also that gcd(m, n) = 1. Then p = m^3 / n^3. Then pn^3 = m^3. Then p | m^3. Since p is prime, we see that p | m. Then pk = m for some integer k. Then p^3 k^3 = m^3. Then pn^3 = p^3 m^3. Then n^3 = p^2 m^3 = p(pm^3). Then p | n^3. Then p | n since p is prime. Since p | m and p | n, we see that p | gcd(m, n). Then p | 1. Then p = 1 which is a contradiction. Thus, cube root of p is irrational.