My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into th… - Niels Henrik Abel
" "My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to do few of the so-called paradoxes. It is really interesting to seek the cause.
In analysis, one is largely occupied by functions which can be expressed as powers. As soon as other powers enter—this, however, is not often the case—then it does not work any more and a number of connected, incorrect theorems arise from false conclusions. I have examined several of them, and been so fortunate as to make this clear. ...I have had to be extremely cautious, for the presumed theorems without strict proof... had taken such a stronghold in me, that I was continually in danger of using them without detailed verification.
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About Niels Henrik Abel
Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for 250 years. He was also an innovator in the field of elliptic functions, discoverer of Abelian functions. Despite his achievements, Abel was largely unrecognized during his lifetime and died at the age of 26.
Also Known As
Alternative Names:
Niels Abel
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Abel, Niels Henrik
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N. H. Abel
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Abel
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Henry Abel
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Additional quotes by Niels Henrik Abel
On the whole, I do not like the French as well as the Germans; the French are extremely reserved toward strangers... Everybody works for himself without concern for others. All want to instruct, and nobody wants to learn. The most absolute egotism reigns everywhere. The only thing the French look for in strangers is the practical; no one can think except himself, he is the only one who can produce anything theoretical. This is the way he thinks and so you can understand it is really difficult to be noticed, particularly for a beginner.
Lety<sup>5</sup> - ay<sup>4</sup> + by<sup>3</sup> - cy<sup>2</sup> + dy - e = 0be the general equation of the fifth degree and suppose that it can be solved algebraically,—i.e., that y can be expressed as a function of the quantities a, b, c, d, and e, composed of radicals. In this case, it is clear that y can be written in the formy = p + p<sub>1</sub>R<sup>1/m</sup> + p<sub>2</sub>R<sup>2/m</sup> +...+ p<sub>m-1</sub>R<sup>(m-1)/m</sup>,m being a prime number, and R, p, p<sub>1</sub>, p<sub>2</sub>, etc. being functions of the same form as y. We can continue in this way until we reach rational functions of a, b, c, d, and e. [Note: main body of proof is excluded]
...we can find y expressed as a rational function of Z, a, b, c, d, and e. Now such a function can always be reduced to the formy = P + R<sup>1/5</sup> + P<sub>2</sub>R<sup>2/5</sup> + P<sub>3</sub>R<sup>3/5</sup> + P<sub>4</sub>R<sup>4/5</sup>, where P, R, P<sub>2</sub>, P<sub>3</sub>, and P<sub>4</sub> are functions or the form p + p<sub>1</sub>S<sup>1/2</sup>, where p, p<sub>1</sub> and S are rational functions of a, b, c, d, and e. From this value of y we obtainR<sup>1/5</sup> = <sup>1</sup>/<sub>5</sub>(y<sub>1</sub> + α<sup>4</sup>y<sub>2</sub> + α<sup>3</sup>y<sub>3</sub> + α<sup>2</sup>y<sub>4</sub> + αy<sub>5</sub>) = (p + p<sub>1</sub>S<sup>1/2</sup>)<sup>1/5</sup>,whereα<sup>4</sup> + α<sup>3</sup> + α<sup>2</sup> + α + 1 = 0.Now the first member has 120 different values, while the second member has only 10; hence y can not have the form that we have found: but we have proved that y must necessarily have this form, if the proposed equation can be solved: hence we conclude that It is impossible to solve the general equation of the fifth degree in terms of radicals.
It follows immediately from this theorem, that it is also impossible to solve the general equations of degrees higher than the fifth, in terms of radicals.
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The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.
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