Abraham de Moivre made significant progress in investigating this question. In today's blog, I will show that we can answer the case of a composite number. If n is a nonprime such that n=rs and the r-th roots of unity can be expressed as radicals and the s-th roots of unity can be expressed as radicals, then it follows that the n-th root of unity can be expressed as radicals.
Let r,s be positive integers
Let μ1, μ2, ..., μr be the r-th roots of unity.
Let ζ1, ζ2, ..., ζs be the s-th roots of unity.
The rs-th roots of unity are of the form:
for i = 1, ..., r and j = 1, ..., s.
(1) Using an earlier result (see Lemma 1, here), we have:
(2) Setting Y = Xr gives us:
(3) Using de Moivre's equation (see Theorem 1, here), we know that:
(4) Combining step #2 and step #3, we have:
Let n be a positive integer.
If for each prime p factor of n, the p-th root of unity is expressible as radicals, then the n-th root of unity is expressible as radicals
(1) If n is a prime number, then the theorem is true by definition. So, for n=2, this theorem holds.
(2) Assume that there exists some number n such that the theorem holds true for all numbers less than n.
(3) If n is a prime, then the theorem is true and we are done. So, we assume that n is not a prime.
(4) Then there exists two numbers r,s such that r≠ 1 and s≠1 and n =rs.
(5) Since by assumption the r-th roots of unity and the s-th roots of unity are expressible by radicals, we can apply Lemma 1 above to conclude that n-th roots of unity must also be expressible by radicals.
The implication of Theorem 2 is that to determine which n-th roots of unity are expressible as radicals, we only need to consider the cases where n is a prime number.
- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001