August 14, 2018 Tuesday
Bedtime Story
Galois group in a Quartic Polynomial
Last night we had got the first taste of
Galois group related to a simple quadratic equation.
Even though not complete, now you will
begin to get the concept of group as applied to the roots of polynomials.
The previous example was a simple quadratic
equation wherein first we had arrived at the roots of the polynomial then
sought ways to relate them algebraically and then consider if that relationship
remains intact if we switch their positions in the algebraic relationship.
We can now advance to a higher degree of
polynomial, namely quartic.
Let us look at the following polynomial.
x4 – 10x2 + 1
The polynomial can also be re-written in a
square form as as follows:
(x2
– 5)2 - 24
This polynomial has four roots which are as
follows:
√2 + √3
√2 - √3
-√2 + √3
-√2 - √3
We can label them as A, B, C and D
respectively in the same manner that we did last time.
Now let us see what kind of valid algebraic
equations can be made with these four roots given above.
Following relationships come to the mind.
AB = -1
AC = 1
A + D = 0
It has to be born in mind that in order to
belong to Galois group, they must preserve the algebraic equation with rational
coefficients.
From this, one then goes one to derive the
Galois group and see if the group is solvable.
How it is done I cannot explain for that
requires in depth knowledge of group mathematics of which I have no
qualification.
If the Galois group of a polynomial
equation is solvable then so too is its corresponding polynomial equation by
radicals.
With this I am sure you will get some sense
of group theory with respect to abstract algebra and polynomials.
Now we need to establish the link of group
theory with the mathematics of symmetry.
One particularly interesting thing that
Galois Theory does when it describes group solvability is that it describes the
symmetries of the roots of the polynomials.
The term ‘symmetries of the roots’ will
need some explanation and only time will tell if I shall be able to explain
that you in future.
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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