
Abel's Theorem in Problems and Solutions (2004) [pdf] - montrose
http://www.maths.ed.ac.uk/~v1ranick/papers/abel.pdf
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sykh
Note that from the preface these notes are based on a half year high school
course. The high school certainly was a specialized math high school. Soviet
education was superb.

I’ve perused the notes and they heavily rely on problem solving to learn the
material. It seems like a Moore method style of exposition which I really
like. Arnold is a master of mathematical writing and teaching.

We no longer teach the formulas for solving cubic and quartic polynomial
equations. Nowadays we just teach linear and quadratic equations. For the most
part every algebraic equation you can solve by hand that we teach in basic
algebra is an equation that can be reduced to a linear equation or a quadratic
equation. It’s amazing how many applied problems can be accomplished by this
reduction. With the rise of computers and easy numerical computations this
isn’t so important but imagine a scenario in which the useful applications all
involved degree 5 or higher polynomials. Would we have progressed much?

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whatshisface
> _imagine a scenario in which the useful applications all involved degree 5
> or higher polynomials. Would we have progressed much?_

You can always approximate a polynomial around a point with a lower-degree
polynomial. They will only diverge farther out. As a result whenever physics
produces a high-degree polynomial we can inspect certain behaviors in a lower
degree. This doesn't help with every case, but it does enough for the
situation that many questions become answerable.

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sykh
Oh definitely. Thinking it about it more it’s clear that part of my comment
was I’ll thought out. I was attempting to wonder if mathematics would have
developed enough theory behind it in a universe in which low degree
polynomials were difficult to solve or in which most applicable problems
involved high degree polynomials. I think the answer is yes.

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DoreenMichele
_I’ll thought out_ = ill thought out

You must be posting from a phone.

#damnauto-corrupt

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sykh
I am! Thanks for catching that mistake.

