Student solves ancient math problem

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Discussion Overview

The discussion revolves around a Dutch student's claim of having solved an ancient mathematical problem related to finding the zero-points of polynomials of any degree. Participants explore the implications of this claim in the context of historical contributions to polynomial equations, particularly focusing on the work of mathematicians like Galois and Abel.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express skepticism about the validity of the student's formula, questioning how it aligns with established mathematical theories, particularly Galois' work.
  • Others highlight the historical context of polynomial equations, noting that no general formula exists for polynomials of degree greater than four, as established by Abel's theorem.
  • A participant mentions the difficulty of understanding the student's paper and suggests it lacks clarity and rigor.
  • Some contributors discuss numerical methods used by graphing calculators to find polynomial roots, raising questions about the relationship between these methods and the existence of a general formula.
  • There are references to Galois' contributions to finite fields and the implications of his work on modern mathematics.
  • A few participants share personal experiences with polynomial equations, indicating a range of understanding and engagement with the topic.
  • One participant proposes that while a general formula may not exist, algorithms could potentially work for a dense subset of polynomials.
  • Another participant mentions their own theorem related to cubic polynomials, suggesting a need for clarity in mathematical terminology.

Areas of Agreement / Disagreement

Participants generally do not reach a consensus on the validity of the student's claim. There are multiple competing views regarding the implications of the student's work, the historical context of polynomial equations, and the clarity of the presented paper.

Contextual Notes

Limitations in understanding the student's paper are noted, with some participants indicating that it may not adequately prove its claims. The discussion also reflects varying levels of familiarity with polynomial theory and historical mathematical concepts.

  • #31
Galois Giant...

Evariste Galois (1811-1832), a French mathematician, proved that it is not possible to solve a 'general' fifth (or higher) degree polynomial equation by radicals. Of course, we can solve particular fifth-degree equations such as x^5 - 1 = 0, but Abel and Galois were able to show that no general 'radical' solution exists.
Galois's Theorem:
An algebraic equation is algebraically solvable if its group is solvable. In order that an irreducible equation of prime degree be solvable by radicals, it is necessary and sufficient that all its roots be rational functions of two roots.
Evariste Galois was 21 years old when he died.[/color]
Abel's Impossibility Theorem:
In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions. This was also shown by Ruffini in 1813 (Wells 1986, p. 59).
Abel-Ruffini theorem:
...states that there is no general solution in radicals to polynomial equations of degree five or higher.
In the modern analysis, the reason that second, third and fourth degree polynomial equations can always be solved by radicals while higher degree equations cannot is nothing but the algebraic fact that the symmetric groups S2, S3 and S4 are solvable groups, while Sn is not solvable for n=>5.
Reference:
Calculus - (Larsen, Hosteller, Edwards) - Fourth Edition - pg.238
http://mathworld.wolfram.com/GaloissTheorem.html
http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html
http://en.wikipedia.org/wiki/Abel-Ruffini_theorem
http://c2.com/cgi/wiki?FermatsLastTheorem
http://proofs-of-fermat-s-little-theorem.wikiverse.org/
http://en.wikipedia.org/wiki/Symmetric_group
 
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