Here comes the best known and most important application of Galois theory: to judge whether an equation is solvable by radicals.One of the most common misconception by folks is that Galois proved equation of degree greater than is not solvable. Since the proof had been given by Abel and Ruffini for near twenty years ago before Galois's work. In fact, Galois proved the sufficient and essential condition, that an equation related to a polynomial is...(read more)

In the problem of counting, the biggest headache is to deduplicate equivalent cases. The simplest one among them is calculating ways of combination of elements out of elements, we first take out elements for permutation, where there're totally ways to do so, then we deduplicate the permutation of these elements and there're ways to combine them. However, going any further leads us to complete chaos,...(read more)

A group action is a binary operation involving a group and a set, resulting in another element in that set, where consecutive group actions are identical to associating the group elements first, and then letting it act on the set element.Consider how one can compute the a term in Fibonacci sequence through , which can be rewritten as ....(read more)

Some of us might have already heard of the famous theorem that algebraic equations of degree over are not solvable by radicals, and by the text of permutation group we've known that is simple when it is on over letters, should these two fact have connection? The answer is, yes.Given an algebraic equation of degree :When it comes to you, what are you thinking...(read more)

Bijections on finite set, which can be proved to be permutations, form a type of group called the permutation group.The study of permutation group is originated from and tied with the solvability by radical of algebraic equations. Ever since the solutions to cubic and quartic equations was published, hundred of mathematicians have floundered in the swamp of finding solution to quintic equations, and none of them succeeded. Lagrange, who summarized and refined former works in depth,...(read more)