The Frobenius automorphism associated to a prime ideal in a Galois extension of number fields was actually developed by Dedekind, who wrote about it (and the associated ramification groups, later found by Hilbert) in a letter to Frobenius on June 8, 1882. Frobenius published this construction in a paper in 1896. Some citations:
Frobenius, Collected Works, Vol. 2, pp. 719--733.
van der Waerden, Modern Algebra, Vol. 1 (Spring 1966), p. 203.
Zassenhaus, Canadian Math. Bulletin 18 (1975), p. 448.