Prove that if K is the root field of any polynomial over Q, and K contains an nth root of any number a, then K contains all the nth roots of unity.
The statement is wrong. A counterexample: K = Q(sqrt(2)). Its a root field of x^2 - 2 and contains 4-th root of a = 4. But clearly K does not contain all 4-th roots of unity, since they are 1, -1, i, -i.
But the statement is correct, when "if x^n - a is irreducible" is added to it.
Prove that if K is the root field of any polynomial over Q, and K contains an nth root of any number a, then K contains all the nth roots of unity.
The statement is wrong. A counterexample: K = Q(sqrt(2)). Its a root field of x^2 - 2 and contains 4-th root of a = 4. But clearly K does not contain all 4-th roots of unity, since they are 1, -1, i, -i.
But the statement is correct, when "if x^n - a is irreducible" is added to it.