Radical extension

In mathematics and more specifically in field theory, a radical extension of a field "K" is an extension of "K" that is obtained by adjoining a sequence of "n"th roots of elements.
Definition.
A simple radical extension is a simple extension "F"/"K" generated by a single element formula_1 satisfying formula_2 for an element "b" of "K". In characteristic "p", we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension. A radical series is a tower formula_3 where each extension formula_4 is a simple radical extension.
Properties.
These three properties show that the class of radical extensions is a distinguished class of field extensions.
Solvability by radicals.
Radical extensions occur naturally when solving polynomial equations in radicals. In fact a solution in radicals is the expression of the solution as an element of a radical series: a polynomial "f" over a field "K" is said to be solvable by radicals if there is a splitting field of "f" over "K" contained in a radical extension of "K".
The Abel–Ruffini theorem states that such a solution by radicals does not exist, in general, for equations of degree at least five. Évariste Galois showed that an equation is solvable in radicals if and only if its Galois group is solvable. The proof is based on the fundamental theorem of Galois theory and the following theorem.
The proof is related to Lagrange resolvents. Let formula_5 be a primitive "n"th root of unity (belonging to "K"). If the extension is generated by formula_1 with formula_7 as a minimal polynomial, the mapping formula_8 induces a "K"-automorphism of the extension that generates the Galois group, showing the "only if" implication. Conversely, if formula_9 is a "K"-automorphism generating the Galois group, and formula_10 is a generator of the extension, let 
The relation formula_12 implies that the product of the conjugates of formula_1 (that is the images of formula_1 by the "K"-automorphisms) belongs to "K", and is equal to the product of formula_15 by the product of the "n"th roots of unit. As the product of the "n"th roots of units is formula_16, this implies that formula_17 and thus that the extension is a radical extension.
It follows from this theorem that a Galois extension may be expressed as a radical series if and only if its Galois group is solvable. This is, in modern terminology, the criterion of solvability by radicals that was provided by Galois. The proof uses the fact that the Galois closure of a simple radical extension of degree "n" is the extension of it by a primitive "n"th root of unity, and that the Galois group of the "n"th roots of unity is cyclic.