I'm Professor Gibbons, and this is the Modern Algeblog. Each week, participants (including me) will contribute commentary, proofs, examples, complaints, reflections (and rotations!) as we progress along our journey.
I'll start! You might remember seeing the quadratic formula at some point in your life. It's kind of nice to have: given a polynomial like \(ax^2 + bx + c\), we can easily find where it crosses the \(x\)-axis -- remember all those interval tests from calculus?!
Turns out there's also a cubic formula for \(ax^3 + bx^2 + cx + d\) (it's not pretty) and even a quartic formula (see below, if you dare) that use only algebra (addition, subtraction, multiplication, division, powers, and \(n\)-th roots).
So what about a quintic formula?
Well, that's what this class is about. As we figure out how to answer that question, we'll encounter the standard topics in a first algebra course: primes, fields, rings, ideals, and groups!
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