Dear P. Karthiksankar
I recommend doing your research about Galois Theory that ends with the exciting result: There is no formula to find the roots of polynomial equation P(x) with degree > 4, in terms of the coefficients of P(x), using only the operations (+, -, x, ÷ ), and the radicals. This will open to you new horizons for doing research.
Best regards
A-3. Beyond the quartic
Answer 3, posted on February 24, 2020
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Nobody knows what the best research topic in algebra is.
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New results on the continuation of the research work on “Beyond the Quartic”
would interest a lot of algebraists, mathematicians, and scientists.
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As always, the first step would be a search for publications done on this subject.
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To get started on this, see:
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Beyond the Quartic Equation
Modern Birkhäuser Classics
R. Bruce King
DOI: 10.1007/978-0-8176-4849-7
ISBN-13: 9780817648367
16 b/w illustrations
150 + viii pages, Springer/Birkhäuser, 1996
https://www.springer.com/us/book/9780817648367
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Quintic polynomial equations:
https://en.wikipedia.org/wiki/Quintic_function
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Sextic polynomial equations:
https://en.wikipedia.org/wiki/Sextic_equation
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With best regards, Jean-Claude
A. 4 Fundamental research
Answer 4. posted 26. Febr. 2020
==========================
Dear P. Karthiksankar,
most research tries to enlarge the building named science at the top. Have a look backwards and think about the roots. There is no safe fundament.
Try to read first — before any decision:
Alberto A. Martinez
negative math [how mathematical rules can be positively bent]
Princeton University Press
ISBN-13: 978-0-691-12309-7
Martinez pointed out the problem, but didn´t find a solution.
After that, you may look at my homepage:
www.mathe-neu.de
You will find proven that
• There is no negative element without inverse addition (subtraction)
• There is no group with different elements and a single combination
• Ring and field are out
• Binomials of the Babyloniens are on circular argument
• Square-root gets reformed
• The square-root of minus one gets a solution, so `i´ is out
After this take your choice. I hope you are able to continnue this work.
Regards, Peter
Choose a topic that you like and know you can handle. If you know how to use as an application that will be best. Others can't decide for you.
I find fascinating areas of algebra like rings and representations that allow you to encroach on what is mathematical analisis including derivatives, Cauchy Riemann conditions and Jacobians. To convert mathematical analysis into algebra.(at least some of it)
Follow geometrical algebra at the same time.
There is one relevant post in my profile.
(Functions over commutative rings)
I think Jean has good intuition, but cannot personally stomach Galois theory. Do not say it is not good, but never suited my taste.
You have to find what you like in algebra.
A. 7 Outruled math
Answer 7, posted 29. Febr. 2020
==========================
Juan Weisz
We shouldn´t point young scientists to the wrong direction.
J. W.: ”...areas of algebra like rings...”
`Ring´ gets defined by two groups. Each group bases on a solitaire (single) combination. The combination of one of the groups is distributive in relation to the combination of the other. The not-distributive group is a complete group. A complete group contains inverse elements.
My reform proofs that there is an inverse combination inside the so called `group´ if there are inverse elements and vice versa.
So no groups with classic defination! No ring, no field!
See my reform.
In a ring some nonzero element do not have inverses within the ring. There is closure under two operarions.
In a group all elements have an inverse within the group. A group has closure under one operation.
Ring is a comon defenition in algebra. Inform yourself.
A. 10 Information to a non-mathematician
Answer 10, posted 29. Febr. 2020
===================================
Juan Weisz
At answer A. 7 I already gave you an information about `ring´.
Look at a good textbook. A ring depents on two operations, as you said. One operation (the non-distributive) is by a (complete) group. The other operation is by a semi-group (no inverse elements). So the elements without invers ones are definately called by the semi-group ( not only `some of the ring´).
But both are groups (complete and semi).
The complete group includes inverse elements by definition ... see the reform
Peter
I do not know if you are right or not, but what I know is that there is no need to define ring in terms of group, or two groups, as you suggest.(You are inventing some kind of new group, for which all elements do not have inverses?)
Usually one operation is similar to addition, and the other to multiplication. The distributive property holds, similar to field.
A ring can be commutative or not . The addition would always be commutative, multiplication sometimes not.
(The case of matrices)
There are special clases of matrices within which multiplication is commutative. I speak about those.
regards, Juan
A. 12
Ring and field ever bases on two groups! That´s the definition of them.
The expressions `addition´ and `multiplication´ are abstacts. The school-algebra (addition: 3 + 4 = 7; multiplication: 3 x 12 = 36) only is a special algebra (one of all).
Each algebra gets defined by a complete variation of equal or different elements in relation to the combination and the results.
a + a = ?
a + b = ?
b + a = ?
b + b = ?
See my reform and get introduced very well!
That is not my definition, nor is it a comon definition as far as I know.
Do not know what your algebras are...I could not understand your last remarks.
Comon algebraic structures are
groups, fields, rings, linear algebras, commutative and non commutative algebras, tensor algebra, matrix algebra, etc.
Spaces in which algebra may develop is also of interest, like vector, Banach, Hilbert, dual, tangent and so on.
Im also interested in dual spaces derived from Hilbert spaces.
A. 14 The basis of communication
Answer 14, posted 01. March 2020
============================
Dear Juan Weisz ,
are `your definition´ on such expressions:
1. (M, o)
2. (M, *)
3. (M, o, *)
[no other definition of any other combination or operation; even on a single element]
Where these equivalences (non-reformed math) are valid?
M = set
o = combination of two elements (operation), result inside the set
* = combination of two elements (operation), result inside the set
If so, we may (re-)start the discussion on common representations.
At math there is no logical difference in the name between the variations of `group´.
Also by the representation: (M, o) or (M, *)
Mathematicians talk about a `complete-group´ as well as about a `semi-group´.
`Complete´ and `semi´ could be distinguished among themselves by the properties of the combination.
Would you agree with Part (1) ?
••••••••••••••
The axioms for to distinguish different properties of combinations (inside the group) are:
Assotiativity
a o (b o c) = (a o b) o c
Commutativity
a o b = b o a
Now math does a logical `jump´, leaves the properties of the combination and extends to a definition / exclusion of special characterized elements (neutral, inverse).
Existence of a neutral-element
There exists an element 0 ∈ M
by which it is valid that 0 o a = a
Existente of an inverse element
For each element a ∈ M exists an inverse element –a
by which it is valid that (–a) o a = 0
Expert opinion (non-reformed math) talks about a `semi-group´ if there only is associativity.
Would you agree with Part (2) ?
••••••••••••••
A ring (M, o, *) is meant if `o´ bases on a complete-group and `*´ bases on a semi-group. There has to be distributivity (left and right) among the combinations.
a * (b o c) = (a * b) o (a * c)
(b o c) * a = (b * a) o (c * a)
Would you agree with Part (3) ?
••••••••••••••
If `o´ is equal to `+´ of school-algebra (addition) must be proven. If `*´ is equal to `x´ of school-algebra (multiplication) must be proven.
Would you agree with Part (4) ?
••••••••••••••
Math confuses.
At normal linguistic level (speaking) an idea added to an idea results at two ideas. In the set of ideas both are equal among themselves. The set defines the only property for its elements (non-different).
Math works like: an idea (wagon) added to an idea (motor) results to a car. First they were ideas; later one focusses on their difference. Result outside the set `wagon´ as well as `motor´.
Regards, Peter
A. 15 / 01. March 2020
It may help if you change the position from descibing to constructing.
a o a = ?1
a o b = ?2
b o a = ?3
b o b = ?4
Is the way of construction. By the questionmarks the combination `o´ gets defined. If a complete set of four answers differ in at least one answer it leads to a different algebra.
Well, I struggle a bit to understand you
Of course there may be confusions in algebra, the same as any other topic.
If you say set, you say nothing about the properties of the elements.
If you say ring, you implicitly know that you have a set of math objects that can combine in two different ways.
Say a+b and a x b
The typical example is the set of integers
.....-3,-2,-1,0,1,2,3,.......
which you can both add and multiply, and always remain within the set (closure)
The inverse of multiplication is division, consider now 1/3
However 1/3 is not contained in the set.
This is what makes it a ring. That the inverse operation cannot be performed and stay within the set.
Group on the other hand has closure under one operation, and the inverse operation is always there.
For example the rotations of a circle about its axis, always has an inverse rotation. This is an infinite or Lie group.
If you had a field, you can always invert non zero elements
example the real numbers. As you see rings have less restrictions
Another example of generally non commutative ring are matrices,
Take the matrix
x y
y x
The determinant is xx-yy, which is zero if x=y or x=-y
Along these lines the matrix cannot be inverted, but the matrices are generally non zero. Therefore you have a ring rather than field However , within this class of matrices, you have the commutative property! Try it out..
Well, here I have been constructing for you (multiply two such matrices)...and get the same kind as a result. This is obvious of course for addition.
If you define for me such terms as semi group or complete group, in terms of these or other examples I may be able to see what you are doing.
A12 comment
It has to be two groups over the same set of objects.
But watch out for the existance of the inverse operation.(or not)Group and ring are contradictory because of this.
A. 18 / 02. March 2020
J. W.: ”It has to be two groups over the same set of objects.”
Do you have knowledge about that for the first time?
It was in A. 14: 1. (M, o); 2. (M, *); 3. (M, o, *); M = set
If one feels able to give advice for a post graduated person, there should be basic knowledge about the subject. I thought you had.
Algebra — at school-level — takes abstracts like `a´ and `b´ for every possibility what number (of amount) could be taken instead. But doesn’t change by the representation. So the `M´ at A. 14 — for to be unambiguous. But answer fourteen wasn’t to learn — only a remembering.
The symbols `a´ and `b´ are generilazations.
At the level of academic knowledge of algebra (constructing every possibility) the symbols of combination `+´ and `x´ get generalized (+ => o / x => *).
If you don’t understand this, it would not be my level in communication / discussion — no teaching basic expert opinion.
By my reform a third level of generalization gets represented. Breaking the separation between the symbols for element and combination.
https://www.researchgate.net/publication/338254411_Logic_of_Formalism_01_-_Subtraction?_sg=vzaq7LxMG3q0Za0qKoIa-wHReACG1f6oYqq85dCIb24OOshRQK-aTKhJ8afcpLPxHYBmwyLYY2n40wIGmAu7z6q4eoUqNaMAKf3a44hb.QujQ8NnY9_GqAFnoxnn6cn5yE5GrsabG5ENizLoG0dWPoeyjkkdpb_cXaFGNJB-vRVUSNvTOW96hN732nte0oQ
https://www.researchgate.net/publication/338302397_Logic_of_Fomalism_02_-_Negation?_sg=vzaq7LxMG3q0Za0qKoIa-wHReACG1f6oYqq85dCIb24OOshRQK-aTKhJ8afcpLPxHYBmwyLYY2n40wIGmAu7z6q4eoUqNaMAKf3a44hb.QujQ8NnY9_GqAFnoxnn6cn5yE5GrsabG5ENizLoG0dWPoeyjkkdpb_cXaFGNJB-vRVUSNvTOW96hN732nte0oQ
That is a bit rude, leave off here. Do not think it is adequate media to push your own stuff around.
With integers a group of addition is fine, but a group of multiplication is not, I explained why. That is why you use ring instead.
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