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.