Re: A bizarre memorial to mathematician Niels Henrik Abel 1802-1829
:lecture:
Not just square roots, e.g., x^3 - 2 = 0.
I was never clear about the relative contributions of Abel and Galois. Did they know each other's work (the timing of their short productive lives overlap). Mathematicians are very bad at their history.
Thanks for the interesting reading assignment -- I did 2-3 years of solid math as an undergraduate, and you can't learn physics without using group theory, both discrete and continuous, but Abel and Galois were both just exotic names to me. Both were from radical families, living in dangerous times, and for various reasons, very short of money. They both died quite young. Abel came first. The hot problem of their time, more or less the "Goedel's decidability problem" of about 1800 was whether or not all fifth-order and higher polynomial equations could be solved by algebraic means, leading to an answer which at worst involved taking the n-th root of something. The problem attracted crackpot solvers, and both Abel and Galois had their papers ignored or misplaced and lost by the giants of the field, such as Gauss or Legendre.
Abel solved the problem (the answer is no), and found a small math journal in which all of his work appeared. Galois was aware of Abel's proof, but Galois' methods were capable of great extension and application to other problems while Abel's proofs have been subject to small improvements and simplifications down to the present day. Galois invented a permutation group of the basic elements from which a polynomial is created and showed that if the group is non-Abelian, there must be some polynomials with roots not expressible as fractional powers. Analysis of the fundamental groups of quartic and simpler polynomials shows they are Abelian, quintic and higher polynomials are generated by groups which are non-Abelian.
Who knew that this distinction between the simplest irrational numbers and those more irrational has such consequences?
Best reference, but still not the whole story: Wikipedia on "Ruffini-Abel Theorem"