August 12, 2018 Sunday
Bedtime Story
Permutation Group or Galois Group
Galois was thoroughly conversant with the
work of Abel and Ruffini on solubility of higher polynomials but wished to
extend the result concerning quintics and higher polynomials still further.
Was there any possible method to know if at
least some of the higher degree polynomials had solutions, and if yes, was
there any decidable method to find it out before hand by just looking at those
equations rather than manually arriving to their solutions?
The answer strangely enough is yes to the
above question.
At the very heart of Galois’ paper lies
this fact – that for any given polynomial equation, there may exist some roots
that are connected to each other by some kinds of algebraic equations.
In other words solubility of polynomials is
possible when there exists certain roots to polynomials and they demonstrate an
algebraic connection.
But even that is not enough for the
complete assessment of Galois Theory as there is one more crucial condition in
addition to the above that needs to be fulfilled.
This condition brings in the concept of
permutation or simple rearrangement to Galois’ paper and hence in today’s mathematics
this subject is studied as permutation group approach to Galois Theory.
This permutation group concept relies on
the fact that some of the roots of a given polynomial can be linked through
certain algebraic equations.
These roots can be claimed to form a
permutation group if and only if these algebraic equations formed by the roots
remain valid even if the roots of polynomials are interchanged in their
position in the algebraic equations.
This means that if you were to interchange
the values of the roots to different places in the algebraic equations, even then
the algebraic equation would remain valid.
In that case, these roots belonging to that
polynomial form a group that is called permutation group or the Galois group of
the polynomial and so finally we can see how group has crept inside the field
of abstract algebra of polynomials.
This was the core essence of Galois paper
and as you can see if this idea is written down in the language of mathematics and
more so in the undecipherable style of Galois would make it virtually incomprehensible
to most, and no wonder it was brushed aside by even the best of the
mathematical minds of his days.
In the history of mathematics pertaining to
abstract algebra and polynomials, Galois was the first person to use the word
‘group’ (in his paper he sued the French word ‘groupe’).
In his original paper, Galois had limited
his idea to algebraic equations whose coefficients were only real numbers.
But as the further developments in abstract
algebra revealed, Galois’ theory could be extended to any mathematical field.
All the verbal explanation about Galois group
will not make much sense unless some specific examples are provided to back it
up.
In the nights to come we shall take up some
simple polynomials and see how we can assign permutation or Galois group to
them.
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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