The set C is inconsistent, as supported by:

Preprint The quantum set Q*

https://www.researchgate.net/publication/355406882/

https://www.researchgate.net/publication/355406639/

https://www.researchgate.net/publication/286722113/

https://www.researchgate.net/publication/236420748/

https://www.researchgate.net/publication/286640675/

https://www.researchgate.net/publication/286640638/

https://www.researchgate.net/publication/243470610/

https://www.researchgate.net/publication/286625459/

and:

[1] Ed Gerck, “Algorithms for Quantum Computation: Derivatives of Discontinuous Functions.” Mathematics 2023, 11, 68. https://doi.org/10.3390/math11010068 , 2023.

[2] Nicolas Gisin, “Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?” Springer Nature, Erkenn86, 1469–1481. https://doi.org/10.1007/s10670-019-00165-8 , 2021.

[3] Yuri Igorevich Ozhigov, “Constructive Physics (Physics Research and Technology).” Nova Science Pub Inc; UK ed., ISBN, 2011.

What seems most inconsistent about C?

Realize there is something genuinely weird:

There’s nothing you can do to distinguish i from -i.

The set ℂ (mathematical complex numbers) is condemned because it uses the set ℝ (mathematical real-numbers), which is condemned (see Gerck 2022, Gisin 2021).

One could think to still use the set G (with rational imaginary numbers), but not only the other inconsistencies apply, but every path must begin and end in a rational number, with no room for imaginary numbers.

In summary, it is time to give up any investigation. It is now an illusion.

But, I don't define nature nor evolution. I am just pointing out "not this way!".

The book "Galois Theory" by Ian Stewart is well regarded, and a third edition is available. It was first printed in 1972.

In it, Ian Stewart gives a proof that a specific quintic equation, x^5 - 6x + 3 = 0, cannot be solved by radicals.

Stewart considers this the strongest statement of the Galois Theory, and by far the most convincing.

The graph of that quintic is given above, and as one can see there are three real roots

The graph above is measurable (cf. Apostol). Therefore, it admits a Taylor representation and a Fourier representation.

Each one is an infinite series with calculated coefficients using rational numbers (the results of derivative and Integral expressions).

There are uncountable other representations that one can write, all using calculated coefficients using rational numbers (the results of derivative and Integral expressions), and the set Q is closed under addition/subtraction.

Therefore, one can write a solution to the quintic considered unsolvable in terms of a calculable expression with rational numbers, which can only add to a rational number. -- a member of the set Q.

Any polynomial with coefficients in the set Q, admits all solutions in the set Q. Nothing is missing. A polynomial of degree n admits at most n solutions.

Possibly, the misuse of the set C confused this issue at the time of Galois and later. There is no such limitation.

The result we report above, confirms that there are uncountable ways to represent numbers.

Not all, however, are physically significant (can be cast in terms of a series with rational numbers as coefficients), as I calculated and published.

For example, mathematical complex-numbers are not measurable. See (Gerck 2022) and (Gisin 2021). This makes the set C not valid. In particular, the p-adic numbers are a concern because their metric is physically untrustworthy.

Quantum mechanics does not use imaginary numbers, as I calculated as early as 1978.

There are other examples, published.

I also affirm, based on my calculations, that a prime number is not a mysterious occurrence in the set Z, but the result of a physical function, measurable therefore. Thus, a prime number is the same on Earth as well as in the star Betelgeuse, which can be measured today.

Nothing speaks deeper to the discredited use of irrational numbers than the current Galois theory, and the putative insolvability of the quintic equation as a function of calculable radicals.

This makes irrational numbers seem quite unnecessary. Not used and not needed .

Mathematics changes, and the quintic polynomial can be solved. Galois lives on, but the theory has to change.

As another example of the behavior of quintic equations, now solvable in the set Q*, consider an equation proposed by IK, as x^5 +x +1 = 0.

The graph is attached. There is only one crossing of the x-axis, which can be calculated in the set Q*, as a rational number.

In conventional theory one gets:

x = (root(-12 root(69) - 100, 3)/6) + (2/(3 root(-12 root(69) - 100, 3))) + (1/3),or

x = −0.754 877 666 246 692 760 049 508 896 358 528 691 894 606 617 772 793 143 989 283 970 646 080 655 128 081 090 738 227 092 842 250 304 ...

At the time of Galois, writes Ian Stewart in "Galois Theory", the natural setting for most mathematical investigations was the complex number system.

However, quantum computing led us to conclude that the set C of mathematical complex numbers is fictional and not calculable with probability 1 [1].

Thus, the basis of the Galois theory is missing. No wonder it's conclusions are to be found faulty in this question.

Thanks for the private messages, I agree that the solution to second-degree equations, and higher, are to be found using the new set Q*.

Dear Ed,

I noticed that your "quintic polynomial" is not a proper quintic polynomial, as it can be factored into a cubic polynomial and a square polynomial as follows:

x^5 + x + 1 = ( x² + x + 1 )·( x³ - x² +1 ) = 0

You are thus not addressing Galois, but Cardano. The first polynomial gives two roots of unity - while the latter polynomial gives the Cardano solution:

xo = [ 1 - 2·cosh(λ/3) ] / 3 = -0.754877666...

So there is no need for Galios here - indeed an overkill, while to simplify the task we use the Guedermann substitution 2·cosh(λ) = 25 from the cubic discriminator. Finally, two other roots follow by using cosh[⅓·(λ±i·π)] in the equation above, where (±) gives complementary roots completing the three roots of the cubic equation ( x³ - x² +1 ).

All the best,

Guðlaugur Kristinn.

Your reference: "As another example of the behavior of quintic equations, now solvable in the set Q*, consider an equation proposed by IK, as x^5 +x +1 = 0."

GKO: Good solution, same answer. But that quintic polynomial, as I said and you quoted, is not mine but proposed by IK.

Can you try the other quintic polynomial, quoted by Ian Stewart; given above? It seems to not have a reducing lower order.

Anyway, mathematical complex numbers are fictional [1], which invalidates Galois theory, and quantum computing hype on non-abelian anyons.

Dear Ed.

Thank you for the question, as a quintic polynomial that cannot be solved by radicals is only slightly different from your quintic as follows:

x^5 - x + 1 = 0

However, this quintic polynomial can be solved using Hypergeometric functions (4F3) in its many disguises, such as Jacobi, Weierstrass, etc, while ultimately being power series of rational terms.

~~~~~

Furthermore, the problem with the complex plane is not its imaginary aspect, but the fact that all 2D spaces are incomplete, due to the fact that an angular motion creates a moment outside the respective 2D space.

This problem is not present in 3D space, as all angular motions create a moment inside that 3D space which renders it physically complete.

One should indeed throw away physical 2D spaces, while technical 2D spaces are nifty tools for studying simple problems and phenomena. One can thus freely project problems down a dimension for further scrutiny while being well aware of the limitations.

All the best,

GKO

Your reference: "As another example of the behavior of quintic equations, now solvable in the set Q*, consider an equation proposed by IK, as x^5 +x +1 = 0."

GKO: Thanks. I consider that the problem with the complex plane is that there is no square-root of a negative nnumber. This use introduces fictions, such an anyons in quantum mechanics. A hopeless tool for quantum computing. Erwin Schrödinger, already on 1926, did not consider imaginary numbers in quantum mechanics. But the me-me opposition seems persistent; albeit with an impotent reach.

The 3D space is not complete physically, and one has to introduce axial vectors and other quantities in angular motions, because in 3D the Gibbs product of two vectors is not a vector. This is solved in 4D and has nothing to do with the error in Galois theory. In games, 4D is used for rotations exclusively. In space navigation, also. Nature is at least 4D.

One can hope that, likewise, complex numbers will be understood to not exist in nature, and thus deprecated. This will open the mind to new solutions, faster and better, and also deprecate Galois theory.

Please see

https://www.preprints.org/manuscript/202305.0936/v1

Thanks Ed, while all these problems go away in 'Natural space' of any dimension, while d = 0 is 'almost' trivial.

When working in 'Natural space' one uses the mathematics generated by the Poisson degeneracy, that manifests for more than one dimension.

One is thus asking Nature about her mathematical content - and to ones surprize, she does have a particle content and hierarchy one can inspect and study for any n-dimensional space of choice.

An extra bonus is almost trivial solutions of n-degree polynomials such our quintic polynomial from yesterday.

All the best,

GKO

Preprint Our Natural Space

Thanks, GKO.

See

https://www.researchgate.net/publication/370755777/

Thanks for asking Sergey P. Klykov

All the best, GKO

Thanks, GKO.

This question is closed. Preprint to follow, including GKO's and private nessages' contributions.

Some people want to remain anonymous, which is a wise way to avoid noise.

Thank you Sergey P. Klykov

4F3( { 1/5, 2/5, 3/5, 4/5 } ; { 2/4, 3/4, 5/4 } ; 55/24 ) = ½

Hypergeometric functions are of special interest in Natural Space, so this is indeed a nice thing to know - and tells a deeper story.

Regarding my n-degree polynomials in Natural Space, I'm adding a new section in Chapter 1 where I go more into the details how nature solves polynomials:

Preprint Our Natural Space

All the best,

Guðlaugur Kristinn

This question served its purposes, and has been closed. I don't define anything; I am powerless against the truth. Nature is the ONLY arbiter. P-adic numbers are physically false in their metric; nature denies them. Thank you all, also for the private messages. See preprint:

Preprint Changing π … while keeping the same value

To reopen the question, as the opposition floats down by the river, indeed it is not only 10^1000 decinal digits factorization in a commercial cellphone that seems to come "out of the left field". We published the numbers, in case someone doubts.

At the time of Galois, writes Ian Stewart in "Galois Theory", the natural setting for most mathematical investigations was the complex number system.

So, IBM and Google cannot be faulted by ... doing the same today!

However, quantum computing (QC) led us to conclude that the set C of mathematical complex numbers is fictional and not calculable with probability 1 [1]. Reference in the QC preprint, in my RG repository.

Thus, the basis of the QC by IBM and Google is missing. No wonder their conclusions are to be found faulty in their QC. They are numerically proving a negative.