abstract-algebra
3 free lessons tagged abstract-algebra across Math. Each one is a short sequence of focused steps with narration and a five-question quiz at the end — take them in any order, no signup required.
Why There Is No Quintic Formula
Quadratics, cubics and quartics all have formulas in radicals. The general fifth-degree equation has none, and not because nobody clever enough has tried. This lesson explains how permuting the roots turns an algebra question into a group question, and why the answer changes at degree five when a group of order 60 refuses to break down.
Subgroups, Cosets, and Lagrange's Theorem
Cosets cut a group into equal-sized blocks, and one counting argument then forces every subgroup's size to divide the group's. This lesson proves that, derives Fermat's little theorem from it as a corollary, and introduces the maps that let you compare two groups and say precisely when they are the same object.
What a Group Is, and Why the Axioms Are So Bare
A group is a set with one operation obeying four rules, and that deliberate poverty is the point: anything proved about groups holds for rotations, permutations, integers, XOR and Rubik's cube at once. This lesson builds the definition from symmetry, works through the symmetries of a square, and shows where groups already sit in code.