While waiting for the royal decree to be issued, in 1824 the published at his own expense his proof of the impossibility of solving algebraically the general equation of the fifth degree, which he hoped would bring him recognition. He sent the pamphlet to Gauss, who dismissed it, failing to recognize that the famous problem had indeed been settled.
Abel spent the winter of 1825-26 with Norwegian friends in Berlin, where he met August Leopold Crelle, civil engineer and self-taught enthusiast of mathematics, who became his close friend and mentor. With Abel’s warm encouragement, Crelle founded the Journal fűr die reine und angewandte Mathematik (“Journal for Pure and Applied Mathematics”), the first volume of which (1826) contains papers by Abel, including a more elaborate version of his work on the quintic equation. Other papers dealt with equation theory, functional equations, integration in finite forms, and problems from theoretical mechanics.
Abel’s early mathematical training had been in the formal school typified by Euler. In Berlin new directions in mathematics stimulated him to do further independent work. Soon distracted socially, however, Abel travelled throughout Europe.
Arriving in Paris in the summer of 1826, he called on the foremost mathematicians and completed a memoir on transcendental functions. In this major work he presented a theory of integrals of algebraic functions, in particular the result know as Abel’s theorem: there is a finite number, or genus, of independent integrals of this nature.