1 F eb 1 99 7 REAL FIELDS AND REPEATED RADICAL EXTENSIONS

1. Introduction. Recall that a field extension F ⊆ L is said to be a radical extension if it is possible to write L = F [α], where α ∈ L is an element with α n ∈ F for some positive integer n. More generally, an extension F ⊆ L is a repeated radical extension if there exist intermediate fields L i with F = L 0 ⊆ L 1 ⊆ · · · ⊆ L r = L and such that each field L i is a radical extension of L i−1 for 0 < i ≤ r. Given a polynomial f (X) over a field F of characteristic zero, let S be a splitting field over F for f. Then as usual, we say that f is solvable by radicals if S is contained in some repeated radical extension of F. A celebrated theorem of Galois asserts that this occurs if and only if the associated Galois group Gal(S/F) is a solvable group. It is well known that intermediate fields of repeated radical extensions need not themselves be repeated radical extensions of the ground field. The solvability of Gal(S/F), therefore, does not guarantee that S is a repeated radical extension of F , and so the phrase " contained in " in the statement of Galois' theorem is essential. For example, take F = Q, the rational numbers, and consider the polynomial f (X) = X 3 − 6X + 2. It is easy to see that f has three real roots, and so we can take S ⊆ R. Of course, the cubic polynomial f is solvable by radicals; we can see this explicitly by calculating that the three roots of f are given by the formula r = α + 2/α, where α runs over the three complex cube roots of the complex number −1 + √ −7. If S were a repeated radical extension of Q, there would have to be some alternative way to express these roots in terms of real radicals. This is impossible, however, since f is easily seen to be irreducible, and it is a classical result that if an irreducible cubic polynomial has three real roots, then these roots definitely are not expressible in terms of real radicals. More generally, we have the following (known) result. (See [1] or Theorem 22.11 of [2]. Also, we include a somewhat simplified …