Sophie Germain had maintained mathematical contacts for some years with such savants as Lagrange and Legendre; but her mathematical career may be said to date from her first letter to C. F. Gauss, 21 November 1804, signed with the pseudonym LeBlanc. Gauss, one year her junior, had published his first great work, the Disquisitiones Arithmeticae, in 1801, and Germain's letter is concerned chiefly with the famous §357. There Gauss, using his theory of periods of the cyclotomic polynomial f(x) = (xp ')j{x 1), and the trigonometrical sums now called Gauss sums (§ 356), showed that the number field generated by the pth roots of unity, ρ an odd prime, contains the quadratic irrationality |/±/?. He expresses that fact by showing that/(x) splits over the quadratic field generated thereby, i.e. can be written in the form 4f= Y2 + pZ2, Y and Ζ being polynomials with integral coefficients. This is among Gauss's deepest contributions to number theory. Sophie Germain, in her letter, generalizes the foregoing equation to g(x) = (xpS ')l(x 1), s any positive integer, obtaining 4g = Y'2±pZ'2. This, however, is quite obvious from Gauss's result and the manifest fact that/ divides g. A.-M. Legendre consecrates the Kiume Partie of Tome 2 of his Theorie des Nombres (second edition, 1827) to an explication of Gauss's theory of the cyclotomic polynomial, and there (No. 512) he discusses some elementary congruence properties of the coefficients of the polynomials Y and Z. Germain, in a note published in Crelle's Journal of 1831, the year of her death, extends Legendre' s observations of the coefficients of her polynomials Y and Z' Most of the details of the investigations described in Sophie Germain's first