Real Fields and Repeated Radical Extensions

Solvability by Real Radicals and Fermat Primes

“…For instance, if K is a real field and f (x) a cubic irreducible polynomial in K [x] with three real roots, none of these can be expressed in terms of real radicals. Lately, a detailed treatment of these questions has been given in [1], [2], [5] and [6].…”

A remark on real radical extensions

“…We shall show that the above question in [4] can be answered in the affirmative by proving Theorem C. Let n be an integer > 2 and f (x) an irreducible polynomial in Q[X] of degree n such that the Galois group of f (x) over Q is the dihedral group D n of order 2n. Assume further that f (x) has at least one real root.…”

“…It has been proved in [4] that this condition is sufficient if p is a Fermat prime (i.e. of the form 1 + a power of 2).…”

“…of the form 1 + a power of 2). In [4] the question was raised whether this property actually characterizes the Fermat primes. In [1] an example is given of an irreducible polynomial in Q of degree 7 and the Frobenius group of order 42 as Galois group having 6 non-real roots and one real root which is not expressible by real radicals.…”