Algebra is the language of modern mathematics. Its goal is to extract formal structures from the usual arithmetic of the integers, allowing for our most basic mathematical intuitions to be abstractly applied to a wide variety of circumstances. The depth and breadth of abstract algebra is surprising, even to seasoned mathematicians. Indeed, to give a few geometric applications, the study of crystals crucially uses group theory, while an early triumph of field theory was the proof of the impossibility of the trisection of an angle with ruler and compass.
Solutions to HW11 have been posted.