Radical extensions and crossed homomorphisms

Abstract

If Ω/F is a Galois extension with Galois G and μ(Ω) denotes the group of roots of unity in Ω, we use the group Z1 (G,μ(Ω)) of crossed homomorphisms to study radical extensions inside Ω. Furthermore, we characterise cubic radical extension, and we provide an example to show that this result can not extended for higher degree extensions.

A remark on real radical extensions

“…We first prove the "if" part of assertion (1) lies in a real radical extension of Q. The unique real root α of f (x) is in the compositum of the fields Q(α i ) and therefore also in the real radical extension of Q.…”

“…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.…”

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].…”