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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeJun 16th 2010

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.

• CommentRowNumber2.
• CommentAuthorIan_Durham
• CommentTimeJun 18th 2010

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.

• CommentRowNumber3.
• CommentAuthorHarry Gindi
• CommentTimeJun 18th 2010
• (edited Jun 18th 2010)

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).

• CommentRowNumber4.
• CommentAuthorIan_Durham
• CommentTimeJun 18th 2010

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.

• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeJun 18th 2010

I made a sly link to solvable group (which is not yet written).

• CommentRowNumber6.
• CommentAuthorDexter Chua
• CommentTimeSep 5th 2016

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.

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeSep 5th 2016

Might it be worth keeping a comment that it may or may not be soluble in char = 2?

• CommentRowNumber8.
• CommentAuthorDexter Chua
• CommentTimeSep 5th 2016

I’ve added the content of #6 to the entry.

• CommentRowNumber9.
• CommentAuthorTobyBartels
• CommentTimeSep 15th 2016
• (edited Sep 15th 2016)

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).