>Everyone who goes through high school is fairly deeply familiar with polynomial equations. A high schooler is not nearly familiar enough with polynomials to u

>Everyone who goes through high school is fairly deeply familiar with polynomial equations. A high schooler is not nearly familiar enough with polynomials to understand the group theory component of Galois Theory. The central observation of Galois theory is that, in certain circumstances, it is possible to permute values without affecting the correctness of any polynomial (with coefficients in a certain field). For example, suppose you have two unknown values, A and B, and knew that the following polynomial equations held: A + B = 0 AB = 1 A^2 = -1 B^2 = -1 There are two possibilities: A=i and B=-i or A=-i and B=i. Further, no matter what polynomials (with rational coefficients) I give you, it is impossible to distinguish these two possibilities. The group in this case is all the possible ways of permuting A and B which do not effect any polynomials (with rational coefficients). There is a lot of work involved in actually developing the theory I summarized above. EDIT: Perhaps more damning for using this approach to teaching group theory is that it is, in general, difficult to actually compute what the Galois group is for a given polynomial.