Polynomials and their Roots - Professor Raymond Flood

We are familiar with the formula for solving a quadratic equation where the highest power of the unknown is a square. The quest for a similar formula for equations where the highest power is three, four five or more led to dramatic changes in how this question was regarded. Powerful techniques in algebra were developed following work by Abel and Galois in the 19th century to show that there is no such formula when there are powers higher than four.
The transcript and downloadable versions of all of the lectures are available from the Gresham College website:
www.gresham.ac.uk/lectures-and...
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[@RichardAlsenz]

This is an excellent presentation. Thank you for the posting.

[@lsbrother]

Puzzled by your remark. If you change the quality - using the cogwheel symbol - the highest quality gives a resolution which seems to me to be as good as anything on KZbin and the audio is perfectly clear.

[@RaysDad]

College was great! I wish I could go back and take a course from this professor.

[@christophersedlak1147]

Thanks.

[@bonbonpony]

08:40 Is there a Fundamental Theorem of Geometry too?

[@hugodegaris4054]

Gutenberg did not invent the printing press. The Chinese had printed books 6 centuries before Gutenberg.

[@MrShnazer]

and newton for calculus.

[@Hythloday71]

Why is it such an amazing result that there be no formula for quintic equations ?

[@jasondowd3099]

Hythloday71 Because it breaks the established pattern. That's always interesting, and is usually -- as in this case -- the result of something deeper to which the pattern break opens the door.

[@BartAlder]

+Hythloday71 It means that given all the operations in algebra, and all the whole numbers then if you are forced to write all equations in terms of just the roots, there are polynomials you cannot ever write which have only whole numbers for all coefficients. These two forms of describing the same object (roots and polynomial expressions) are not equivalently all-powerful. This is true for degree 5 and higher. The *exact opposite* is true for degree 4 and lower.

[@bonbonpony]

It's not that there is _no_ such formula at all - it's just that this formula cannot be based on simple arithmetic operations we all know, such as addition, subtraction, multiplication, division, exponentiation or root extraction. And this is a _huge_ difference - something we're usually not being told about :P Here's how you can look at it: You are given a game (a set of rules you can follow) in which you can win by following these rules in a certain way (solving the equation by zeroing it out). For some simple versions of this game (the equations of degree less than 5), the winning strategy (the solution) can be found relatively easily. But to win the 5th level of the game and higher, it turns out that you would have to "cheat" by making an "illegal" move that takes your piece outside the board for a while and magically transporting you to some other place of the board. You have to "transcend" the rules of the game if you want to always win. And this "transcendence" is what makes this discovery special, because it opens your eyes to some new possibilities in mathematics - it shows that there are some magical moves (algebraic operations) that "transcend" the usual arithmetic ones, and that there are also new types of numbers associated with them (so called "transcendental numbers"), which are "unreachable" by algebraic formulae involving just basic arithmetic operations, so you cannot come up with a nice formula for them.

[@jdm3656]

Because there is no intuitive way to grasp that. Think about it: there are infinitely many natural numbers (1, 2, 3, 4, 5 etc); and general polynomials from degree 1 to 4 can be solved via algebraic formulae... So why not 5 and upwards?? Where and why does this “departure” happen?? 🤔🤔🤔 Absolutely fascinating, at least as far as I’m concerned.

[@XiaosChannel]

ahh the interlacing

[@Shockszzbyyous]

purrrr

[@LifeIsBeautiful-ki9ky]

I will solve any quintic equation.

[@oxm18]

Why did the Italians start with all that polynomials!!

[@Tadesan]

mussolini

[@bonbonpony]

I guess it's because of the famous universities that has been opened in Italy, making it a scientific basin of Europe. As for the polynomials, it all started from Luca Pacioli's lectures in which he claimed that solving equations of degree 3 or more is as impossible as squaring the circle. It was taken as sort of a challenge by some mathematicians these days, and they all started cracking their heads to find a solution to this famous problem. Especially after Scipione del Ferro actually found such a solution to one special case of such an equation of degree 3, and passed it to his student, Antonio Maria Fior, who then started bragging about that and challenging other mathematicians for math duels. One could win a serious amount of money on such duels back then, so finding a solution to such problem could be very beneficial ;J

[@AlfaHanen1]

Thanks for that, We have to thank Muslims for Algebra :)

[@bonbonpony]

It's sad that the Muslims of today aren't as wise and into mathematics, science and culture as they were all those centuries ago :( Now they devastate ancient monuments, burn libraries, push their bastardized version of the original religion down people's throats and kill people for whatever stupid reasons they could come up with :/ Where's Allah when he's needed the most? :/