KUMMER, Ernst Eduard (1810-1893). "Theorie der idealen Primfaktoren der complexen Zahlen, welche aus den Wurzeln der Gleichung w/n = 1 gebildet sind, wenn n eine zusammengesetzte Zahl ist," [With:] "Einige Sätze über die aus den Wurzeln der Gleichung (Alpha hoch-Lambda) = 1 gebildeten complexen Zahlen, für den Fall, dass die Klassenanzahl durch (Lambda) theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermatschen Lehrsatzes," [And:] "Über die allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist," [And:] "Zwei neue Beweise der allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist," all from Abhandlungen der Königl. Akademie der Wissenschaften zu Berlin, issues 1856, 1857, 1859, and 1861. Berlin, 1857-1862.

First editions of four major papers on Kummer’s theory of ideal prime factors in relation to Fermat’s Last Theorem. Kummer, a German mathematician and teacher, had realized that attempts to prove Fermat’s Last Theorem broke down because the unique factorization of integers did not extend to other rings of complex numbers; thus he sought to restore the uniqueness of factorization by introducing ‘ideal’ numbers. Not only have his papers been fundamental in work relating to Fermat’s Last Theorem, but the concept of an ideal enabled the development of ring theory and much of abstract algebra.

The Paris Academy of Sciences awarded Kummer the Grand Prize in 1857 for the second paper. In fact, the prize of 3000 francs was originally offered for a solution to Fermat’s Last Theorem—but when no solution was forthcoming, it was given to Kummer (even though he had not submitted an entry for the Prize). The third paper develops general reciprocity theory and prime factorization, and the last paper presents two new proofs of the general reciprocity theory.

Four volumes, quarto (c. 287 x 233mm). Original brown printed wrappers (occasional chipping, hinges just starting on 2 vols).