August 15, 2018 Wednesday
Bedtime Story
Development of Galois Group Theory
While Galois was the originator of the idea
of permutation group in relation to polynomials it was not recognized so during
his lifetime.
His published works were largely ignored
and the notes that he jotted don the night before the duel just barely survived
thanks to his friend Auguste Chevalier.
It was in his last letter and the second of
the three attached manuscripts given to Chevalier where Galois had outlined the
ideas of linear groups over finite fields.
He did so taking the remark of Poisson
seriously who had just few years earlier declared his work to be
‘incomprehensible’ and argument to be ‘neither sufficiently clear nor
sufficiently developed to allow us to judge its rigor’.
Yet Poisson had also rejected the paper
with a positive note saying, “We would then suggest that the author should
publish the whole of his work in order to form a definite opinion.”
It was on account of this advice that he
worked feverishly the whole night in the jail before the duel, knowing very
well that that this could possibly be his last chance to pen his thoughts down
in a more decipherable manner.
Then after that fateful day of May 30, 1832
it was left to the other men of mathematics to understand Galois’ ideas and elaborate
on it.
Group theory after Galois was developed largely
by four European minds.
They were Augustin-Louis Cauchy (France),
Arthur Cayley (Britain), Christian Felix Klein (Germany) and Sophus Lie
(Norway), all from different European countries.
Augustin-Louis Cauchy – French
mathematician and a contemporary of Galois who actually for little understood
reason as a referee refused to publish the two papers submitted by Galois.
Yet he must have understood the fundamental
argument of Galois’ paper since he himself made great advancement in the study
of permutation groups of abstract algebra later on.
Arthur Cayley – British mathematician who
gave the modern definition of groups.
He defined the word ‘group’ in its modern
sense.
Previous to Cayley the word group was meant
to be understood as permutation group.
Cayley defined group as a binary operation
that satisfies certain defined laws.
Christian Felix Klein – German
mathematician in 1872 proposed his Erlangen program (named after University of
Erlangen-Nuremberg located in the cities of Erlangen and Nuremberg in the state
of Bavaria where Klein worked).
Erlangen had appointed Klein as a professor
of mathematics in the year of 1872; Klein was just 23.
He was probably the first person to
systematically link group theory to geometry and making a claim that the group
theory was the most useful way of understanding geometry.
We shall continue with the story of further
advancement of Galois Group theory by other mathematicians – both his
contemporary and future - in the nights to come.
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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