TITLE: Ideas for introducing Galois theory to advanced high school students
QUESTION [48 upvotes]: Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (factorisation via Horner, polynomial division, discriminant and Vieta's formulas for quadratic equations).
I am struggling to find a coherent approach that makes use of their current understanding, e.g., whether to give an intro on group theory, or to describe permutation operations on roots. This is all intended for a short course introduction and not an extended program. Any advice would be quite helpful.

REPLY [2 votes]: Please take a look at Math Girls 5: Galois Theory, which I translated and was published just last month. If you aren't familiar with the series, these books, originally published in Japan, are novels in that they present fictional characters (high school students who explore various topics in math for fun) and have a thin veneer of plot, but most of the content is mathematics.
MG5 isn't a comprehensive presentation of Galois Theory, but rather focuses on Galois' original paper regarding solutions of equations. The bulk of that comes in the final chapter, with preceding chapters introducing permutations, symmetric polynomials, and other requisite information. There's a lengthy fan review of the book on Amazon.com giving a much more detailed description of the content, if you're interested. You can also use Amazon's "Look Inside" feature to browse the table of contents.
Because these books are written with the aim of being accessible to motivated high school students, perhaps they will give you some ideas regarding how to present this challenging content?