August 08, 2018 Wednesday
Bedtime Story
How
This 1830 memoir of Évariste Galois on the
solvability (rather the unsolvability) of polynomials by radical goes by the
title, ‘Memoire sur les conditions de resolubilite des equations par radicaux’ went
unpublished for sixteen years.
It
was only in 1846 it was published by the French mathematician Joseph Liouville
who may not have been the first one to read it to read and certainly was the
first to understand the true importance of this paper.
The paper was published in the French
journal that Liouville himself had founded in 1836 ‘Journal de Mathematiques
Pures et Appliquees.’
Anybody who is even little bit interested
in the lives of European mathematicians would be aware of the fact that Galois
had died in 1832 in the duel and at the time of his death was just 20 years old.
And hence it was left up to the other
mathematicians who were to follow him to read his work, figure it out and recognize
his genius.
Even before Liouville published Galois’
paper in 1846, three years before 1843 he had announced the results of Galois’
findings in a lecture.
Liouville clearly stated in his publication
of Galois’ paper that while Abel’s proof is applicable for general polynomials,
Galois actually defined concrete quintic polynomial whose roots cannot be
expressed in radicals from its coefficients.
So what had Galois come up with?
Galois had found that if r1, r2, r3,…rn are
the n roots of a polynomial equation, there is always a group of permutations
of the r’s such that:
One – Every function of the roots
invariable by the substitutions of the group is rationally known, and
Two- Conversely, every rationally
determinable function of the roots is invariant under the substitutions of the
group.
This does not seem to make much sense, does
it?
Not to me at least.
So let us try to figure out in an
unsophisticated way what Galois did.
You have to bear in mind that real science
and mathematics are difficult subjects and there is a limit to how much one can
simplify.
So let us address the question once again
that fascinated Galois, Abel, Ruffini and Lagrange among others:
Why is there no general formula for the
roots of fifth or higher degree polynomial equations in terms of the
coefficients of the polynomial using only the five algebraic operations of
addition, subtraction, multiplication, division and radicals (square roots,
cube roots and so on)?
When somebody is bothered by such a useless
question you know for sure that such a person is bound to be a mathematician
even if that specific person fails to arrive at the answer to self-raised
questions.
This perhaps also will help you understand
what mathematics is about rather than what we have been forced-fed in our
elementary and high schools.
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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