A simple radical extension is a simple extensionF/K generated by a single element satisfying for an element b of K. In characteristicp, 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 where each extension is a simple radical extension.
Properties
If E is a radical extension of F and F is a radical extension of K then E is a radical extension of K.
If E and F are radical extensions of K in a common overfield C, then the compositumEF is a radical extension of K.
If E is a radical extension of F and E > K > F then E is a radical extension of K.
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 polynomialf 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 be a primitive nth root of unity. If the extension is generated by with as a minimal polynomial, the mapping induces a K-automorphism of the extension that generates the Galois group, showing the "only if" implication. Conversely, if is a K-automorphism generating the Galois group, and is a generator of the extension, let The relation implies that the product of the conjugates of belongs toK, and is equal to the product of by the product of the nth roots of unit. As the product of the nth roots of units is, this implies that 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 nth root of unity, and that the Galois group of the nth roots of unity is cyclic.