$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\Aut(F)$ is generated by the Frobenius automorphism $\phi:F\to F$, $\phi:x\mapsto x^p$.
Question 1: Who was the first to prove this result and when? (Galois, Jordan, Frobenius, Burnside, Moore, Artin?)
I have found the paper "A history of Galois fields" by Frederic Brechenmacher but unfortunately could not locate a precise answer to my question (maybe I have overlooked it, because the paper is rather long). Also a more general question on the entire history of Galois theory was asked here, but also without precise answers.
Known proofs of the cyclicity of the automorphism group of a finite field use another classical result (saying that the multiplicative group of a finite field is cyclic). Many proofs of this classical result (including my own proof) are collected in this MO-post. Now the same historical question arises:
Question 2: Who was the first to prove that the multiplicative group of a finite field is cyclic? (Galois, Jordan, Frobenius, Burnside, Moore?)
