August 05, 2018 Sunday
Bedtime Story
Niels Abel Finally Completes the Unfinished Work of Paolo Ruffini
In the last night’s bedtime story I had stated
that in the story of insolubility of polynomials higher than quintics, the
first person to come with the thought that there exists no general solution in
radicals that applies to all equations of a given degree higher than 4 was
neither Galois, nor Lagrange, nor Abel nor Ruffini but it was our good old
Gauss.
No matter how much you want to escape him,
Gauss eventually and inevitably manages to pop up almost in any subject on
mathematics.
Later in a thesis of his that he wrote a
year later Gauss further adds to this topic:
“After the labors of many geometers left
little hope of ever arriving at the resolution of the general equation
algebraically, it appears more and more likely that this resolution is
impossible and contradictory.
Perhaps it will not be so difficult to
prove, with all rigor, the impossibility of for the fifth degree.
I shall set forth my investigations of this
at greater length at another place.”
That never happened as Gauss never published
anything else on this subject.
Studying the paper of Lagrange, Ruffini
came to his own conclusion that there had to exist a strong connection between
permutations and the solvability of algebraic equations; in fact he established
this.
Following this he made an assertion that
greatly unsettled many mathematicians of his time: that quintic and
higher-order equations cannot be solved by radicals.
Though he made such a bold assertion in
1799 landmark paper titled ‘General theory of equations, in which the algebraic
solutions of general equations of higher degree than four is proven
impossible’, he could never prove it.
It was an incomplete proof in spite of
being a major advancement.
Some 25 years later the great Norwegian
Niels Abel in 1824 worked out its proof and hence in honors to them this
assertion and now a theorem goes by the name of Abel-Ruffini theorem sometimes
also known as Abel’s impossibility theorem.
The theorem states that there is no
algebraic solution, meaning there is no solution in radicals, to the general
polynomial equations of degree five or higher with arbitrary coefficients.
It must be born in mind that this theorem
does not affirm that some higher-degree polynomial equations have no solutions.
Many novices may get this idea which is
totally wrong as the fundamental theorem of algebra that deals with polynomials
says just the opposite:
Every non-constant polynomial equation in
one unknown, with real or complex coefficients, has at least one complex number
as a solution.
In computer science and mathematics there
exists root-finding algorithms which can find out numerical approximations of
the roots of a polynomial equation of any degree.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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