The algebraic numbers that are now commonly called "Gauss sums" were studied in more general form than that introduced in Gauss's Disquisitiones by Lagrange [1]. In that same work, Lagrange shows how to generate an abelian extension of degree n by adjoining an nth root after, if necessary, adjoining the nth roots of unity. These generators were later called "Kummer generators". Jacobi sums, which are closely related to Gauss sums, were studied earlier than Jacobi by Gauss and Cauchy.

Finally, a story best recounted by Weil [2]: "For reference, we recall that the Gauss sums appear among the local constant factors in the functional equations of the $L$ functions; these factors are also called "nombres radiciels" ("root-numbers", "Wurzelzahlen"), undoubtedly because of Hilbert, who a had a sort of genius for bad terminology, where it would have been sensible to name "Wurzelzahl" that which before him had been named a "Lagrange resolvent" , and "Lagrangian Wurzelzahl" that which here has been called a Gauss sum".

[1] Lagrange, Reflexions sur la resolution algebrique des equations, Nouveaux Mem. de l'Acad. R. des Sc. et B.-L. de Berlin, 1770-1771 = Oeuvres, vol. III, p. 332;

[2] Weil, La Cyclotomie Jadis et Naguere.