August 13, 2018 Monday
Bedtime Story
Galois Group in a Quadratic Polynomial
Tonight we shall go totally algebraical and
consider some specific albeit simple examples of permutation groups or Galois
groups in a polynomial.
The simplest case that would suffice as an
introductory example would be a classical quadratic equation that all of us
would be familiar with.
Consider x2 - 4x + 1 = 0
You can solve it with the well known
solution to the quadratic formula which will give you its two roots which are
2 + √3 and 2 - √3
We will label the solution or the roots as
A and B
Let A = 2 + √3
Let B = 2 - √3
Now the next step is to think of some
possible algebraic equations which would be satisfied with these two roots.
In this particular case, it is not two
difficult to come up with two algebraical equations, one of which is the sum of
A and B and the other its product.
So we get the following two equations.
A + B = 4 and
AB = 1
You can verify if they are true by solving
them up yourselves using a pencil and a paper (in case you really use them
these days).
Now we need to permutate or interchange the
roots A and B and see if we can arrive at a true algebraic statement.
Here for instance if A + B = 4, then so
also is B + A = 4
Similarly, if AB = 1, then BA is also equal
to 1.
Now you perhaps get why they were called
permutation groups.
Can these algebraic relations be further
extended?
Yes, and very much so.
The truth is that (even though it is hard
to mentally visualize it) this holds true for every possible algebraic relation
between A and B as long as the coefficients are rational.
Now the roots A and B can be placed in two
kinds of permutation groups.
One permutation group is the identity
permutation group and the other is a transposition permutation group.
The terms ‘identity’ and ‘transposition’
here are being used as applied in the mathematics of permutation.
If I will have to explain the mathematics
of these two terms, I will have to digress far from our subject into the
mathematics of permutations.
So for now just simply accept it.
We shall continue with more complex or
polynomials of higher order in the nights to come to understand the idea of
Galois group in abstract algebra.
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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