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I know that it’s pretty elementary, but sometimes teaching algebra makes you think of things, so I preserved some observations (mostly not my own) on the quadratic formula the other day.
Actually, I think this might be a cool place to point out that attempting to find similarly simple solutions to higher-ordered polynomials is eventually what led to the invention/discovery of group theory (Mario Livio’s got a a really great book about the whole history of that). Plus, my students often find it really cool that, since group theory is the language of symmetry, there’s a link between polynomials and symmetry.
Gauss was using group theory for a significant amount of time before the work of Abel/Ruffini and Galois. What they discovered was the notion of solvability and the use of groups to probe the symmetries of finite separable field extensions (or in the classical formulation, the symmetries of polynomials).
What they discovered was the notion of solvability and the use of groups to probe the symmetries of field extensions (or in the classical formulation, the symmetries of polynomials).
Still, I think it’s an interesting connection and it’s certainly an interesting historical point since a lot of the ideas developed there filtered into the broader theory along the way.
I made a sly link to solvable group (which is not yet written).
Removed the comment that the $a x^2 + b x + c = 0$ is soluble in characteristic $2$ only if $b = 0$, since $x^2 + x$ is a counterexample ($0$ is always a root). The sly link is thus no longer present.
Might it be worth keeping a comment that it may or may not be soluble in char = 2?
I’ve added the content of #6 to the entry.
Thanks, Dexter. David, I've done as you suggested. Zoran wrote solvable group back in 2011 (and I've since edited it), but now I made the link to solvable polynomial (which I just wrote).
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