August 07, 2018 Tuesday
Bedtime Story
Shortcoming of Abel-Ruffini Theorem
Last night we had seen that Abel had read
the proof of Ruffini regarding unsolvability of higher polynomials but was left
a bit unimpressed with it.
Abel was extremely generous in not using the
word incomplete or ‘containing an error’ for Ruffini’s proof for the good
reason that even what Ruffini had achieved was a landmark in itself.
I shall not go into the incompleteness of
the Ruffini’s proof as it is a technical subject and would be of interest only
to pure mathematicians than the general reader.
If the proof of Ruffini was incomplete, the
proof worked out by Abel had its shortcomings in a different way; while his
theorem proved that the general quintic and higher polynomial equations were
unsolvable by radicals, it did not provide for the necessary and sufficient
conditions as to which quintic and higher polynomials are unsolvable by
radicals.
After all, he was obviously well aware that
there certainly existed solutions in radicals to some polynomials of degree 5
and higher than 5.
The question was whether there exists a
method of knowing beforehand that a certain given polynomial of higher degree
would be solvable in terms of radicals?
It is not the case that Abel was unaware of
this shortcoming; He began to work on this after publishing his proof in 1824
which he had to do at his own cost.
Publishing the paper was an expense affair
for Abel for he lived a life of poverty and just five years later he died from
tuberculosis in 1829.
He was only 26.
What was left incomplete by Ruffini and
Abel was completed by Galois the very next year after the death of Abel.
In 1830 the very young Galois (he was just
18 and destined to die very young like Abel) submitted to Joseph Fourier, the
secretary French Academy of Sciences a memoir which described the conditions
necessary for the solvability of polynomials by radicals.
Galois was very well aware of the proofs of
both Ruffini and Abel as he wrote in the paper:
“It is a common truth, today, that the general
equation of degree greater than 4 cannot be solved by radicals…this truth has
become common (be hearsay) despite the fact that geometers have ignored the
proofs of Abel and Ruffini…”
He was motivated to do so by Cauchy.
This 1830 memoir on the solvability of
polynomials by the radical goes by the title, ‘Memoire sur les conditions de
resolubilite des equations par radicaux’ and surprisingly was rejected in 1831.
I am not translating the French title as
its English equivalent seems obvious enough with bare little application of one’s
mind.
As always happens with such geniuses who
come way ahead of their times they and their work are ignored because of lack
of comprehension by the existing specialist contemporaries.
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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