The existence of the Frobenius endomorphism probably goes back to Euler's proof of Fermat's little theorem. But why is it named after Frobenius? Who gave it this name? When was it first stated in full generality? How did people refer to this concept before the language of ring homomorphisms?



History of the Frobenius Endomorphism?
This is traced in Hasse ([1967](//ams.org/mathscinet-getitem?mr=218330)) and Hawkins ([2013](//ams.org/mathscinet-getitem?mr=3099749)), who writes on p. 326: 



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> According to Miyake ([1989](//ams.org/mathscinet-getitem?mr=1018713), p. 347), Hasse introduced the term “Frobenius substitution” in ([1926](//zbmath.org/?q=an:52.0150.19)-[1930](//zbmath.org/?q=an:56.0165.01)), apparently unaware that Dedekind had discovered it independently and probably earlier.
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