quintic

Is it just me, or did they really just fit a quintic curve to theirs "Fraudsters are Sneaky" plot?

Take Galois proof of the insolvablity of the quintic.

Who would have thought that the dodecahedron is involved with solving the quintic?

Unsolvability of the general quintic pared down to just 6 pages.

I only mention because I also recommended this book a month or so ago on the post about the quintic [1].

You can reword this as saying that none of the infinitely many candidate solutions for quintics work.

If you are OK with that situation, why not allow a quintic formula that involves lambda expressions?

Abel's theorem says there is no general formula to solve the quintic and higher order polynomials.

Right, but the dodecahedron is a modular curve, and I'm guessing this is how Hermite came up with the solution to the quintic using modular functions in 1858.

If one didn’t know the original order was cubic, and if one tried to interpolate over the 5 points to fit a quintic, that would be an incorrect reconstruction.

Even if you had mathematica-grade program implementations of all mathematical objects and properties, you can't get over things like "you can't solve the quintic this way, dummy!

So while Western mathematicians were struggling in vain to find a way to double a cube using a compass and straightedge or to derive a quintic formula, the Chinese mathematicians were satisfied with an algorithm that found useful approximations.

So then of course the final proof of the unsolvability by radicals of a general quintic just feels like word and symbol soup that you have to learn by rote, because how could anyone possibly have derived it in any sort of deliberate or systematic way?

What makes radicals so special?Remember, all the Abel-Ruffini theorem says is that there is no quintic formula involving only arithmetic and radicals -- it leaves open the possibility of another kind of formula for the quintic.

What started my journey was looking into the insolubility of the general quintic polynomial equation, something you learn in high school as a random factoid but has a lot of depth.- Studying from small textbooks that I can fit in a backpack, so I can "make progress" during my commute.