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下記 Speech of Prof. Dr. P. L. M. Sylow より抜粋
file:///C:/Users/seta/AppData/Local/Temp/1902_Sylow.html
Abel’s favorite theme was, however, the theory of algebraic equations. Also here he had worked from first principles.
Gauss and Cauchy had given proofs of the Fundamental Theorem of Algebra, to which later mathematicians only have had little to add.
Gauss had in addition exhaustively treated the equations connected to the problem of circular division into equal parts,
Abel proved the impossibility of a general method of solving equations of degree higher than four by radicals, and thereby brought the theory to a rather new level.
He then set out to determine those equations which can be solved in that way,
and discovered the most important general results in this new field. But death prevented him to present his findings,
so that his successor Galois, one of the most outstanding minds of the past century, had to redo those discoveries once again:
because Galois died before the collected works of Abel were published for the first time. Furthermore it was Abel, who first taught the mathematicians to use the auxiliary tool,
which now has been named the Galois resolvent; Galois himself expressively announced that the idea was Abel’s. Finally Abel learned to solve that class of equations, which now bears his name.
His other theories gave him rich opportunity to apply this discovery and show its worth.
But much more important is that the two latter discoveries: Galois’ Resolvent and the theory of Abelian equations, were the two most important tools for Galois,
when he gave the theory of equations its final form and thereby gave the foundation for the rise of our contemporary theory of groups.