Sunday, September 27, 2009

Galois' Memoir on the Solvability of Equations

Evariste Galois died in 1832 when he was just 20. He had made many attempts to gain attention to his theory of equations but each time had failed.

In 1829 when he was 17, he presented his findings on the the solvability of equations to the Paris Academy. Augustin-Louis Cauchy was appointed referee. There is a popular legend that Cauchy did not appreciate the work or somehow lost it but this does not seem to be the case. It is believed that Cauchy presented detailed comments to Galois and suggested that he resubmit his work. For more information on this, see this article. It was at this time that tragedy struck and Galois' father committed suicide. Galois' submission of his work was delayed.

It is believed that being given encouragement from Cauchy, he rewrote his memoir. He resubmitted his memoir to the Paris Academy in 1830. Jean-Baptiste Joseph Fourier was appointed referee. Unfortunately, Fourier's health took a turn for the worse and he died a few weeks after being appointed referee. No assessment was made and the paper was lost among Fourier's other papers.

In 1831, Galois was asked by Simeon Poisson to resubmit his memoir. After reviewing, Poisson rejected the paper as not fully developed. It is interested to note that Poisson was quite impressed by the quality of work but was not able to verify that they were correct.

Here is what Poisson wrote (see reference below for source):
We have made ever effort to comprehend M. Galois's proof. His arguments are neither sufficiently clear nor developed for us to judge their rigor, and we are not in a position to even give an idea of them in this report...

The author claims that the proposition which is the subject of his memoir is part of a general theory rich in application. Often, different parts of a theory are mutually clarifying, and it is easier to understand them together than in isolation. One should rather wait for the author to publish his work in entirety before forming a definite opinion.

Galois took all this very hard as can be imagined. There is another myth at this point that on the night before Galois' duel where he would die, he wrote up his theory in a single night. While he did write a letter to his friend about the nature of his work, it is clear that he had been working on the ideas since he was 17 and the fullest expression of them up to this point had been the paper that was rejected by Poisson.

Galois would die on May 30, 1832. His funeral was on June 2. His good friend Auguste Chevalier and his brother Alfred collected all his papers in the effort to get notice for his work. They were submitted to the most famous mathematicians of the day: Carl Gauss, Jacobi, and others. By 1843, they had made their way to Joseph Liouville.

Finally, they got the attention they deserved when Liouville published Galois's memoir with additional comments in 1846.

In the next set of blogs, I will review the famous memoir by Galois that had been rejected by Poisson and which was modified the night before his death. For an English translation of Galois' memoir, see Harold Edwards' Galois Theory.


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