This theorem forms the basis for the later theory of Abelian integrals and Abelian functions. Abel was accepted with restrained civility in Paris, for his work was still unknown. He submitted his memoir for presentation to the Academy of Sciences, hoping that it would establish his reputation; but he waited in vain. Before leaving Paris thinking he had a persistent cold, Abel consulted a physician, who informed him he had tuberculosis.
Abel returned to Norway heavily in debt. He subsisted by tutoring, by receiving a small grant from the university, and in 1828, by accepting a substitute teaching position. His poverty and ill health did not decrease his production; he wrote a great number of papers, principally on equation theory and elliptic functions. Among them are the theory of the Abelian equations with Abelian groups. He rapidly developed the theory of elliptic functions in competition with Karl Gustav Jacobi.
By this time Abel’s fame had spread to all mathematical centers, and strong efforts were made to secure a suitable position for him by a group from the French academy, who addressed Bernadotte, the king of Norway-Sweden; Crelle worked to secure a professorship for him in Berlin.
In the fall of 1828, Abel became seriously ill, and his condition deteriorated on a sled trip at Christmas time to visit his fiancée at Froland, where he died on April 6, 1829. The French Academy of Science published this memoirs in 1841.
