Editing Abel–Ruffini theorem (section)

===Algebraic solutions and field theory===
An algebraic solution of a polynomial equation is an [[expression (mathematics)|expression]] involving the four basic [[arithmetic operations]] (addition, subtraction, multiplication, and division), and [[root extraction]]s. Such an expression may be viewed as the description of a computation that starts from the coefficients of the equation to be solved and proceeds by computing some numbers, one after the other.

At each step of the computation, one may consider the smallest [[field (mathematics)|field]] that contains all numbers that have been computed so far. This field is changed only for the steps involving the computation of an [[nth root|{{mvar|n}}th]] root.

So, an algebraic solution produces a sequence
:<math>F_0\subseteq F_1\subseteq \cdots \subseteq F_k</math>
of fields, and elements <math>x_i\in F_i</math> such that 
<math>F_i=F_{i-1}(x_i)</math> for <math>i=1,\ldots, k,</math> with <math>x_i^{n_i}\in F_{i-1}</math> for some integer <math>n_i>1.</math> An algebraic solution of the initial polynomial equation exists if and only if there exists such a sequence of fields such that <math>F_k</math> contains a solution.

For having [[normal extension]]s, which are fundamental for the theory, one must refine the sequence of fields as follows. If <math>F_{i-1}</math> does not contain all <math>n_i</math>-th [[roots of unity]], one introduces the field <math>K_i</math> that extends <math>F_{i-1}</math> by a [[primitive root of unity]], and one redefines <math>F_i</math> as <math>K_i(x_i).</math>

So, if one starts from a solution in terms of radicals, one gets an increasing sequence of fields such that the last one contains the solution, and each is a normal extension of the preceding one with a [[Galois group]] that is [[cyclic group|cyclic]].

Conversely, if one has such a sequence of fields, the equation is solvable in terms of radicals. For proving this, it suffices to prove that a normal extension with a cyclic Galois group can be built from a succession of [[radical extension]]s.