Any cyclic extension over fields with characteristic $p$ with degree $p$ that it is the splitting field of a polynomial of the form $X^p-X-a$

Question

I found this statement in Milne's Fields and Galois Theory (page 72):

More precisely, it is the last line I would like to prove. Could someone help me to prove that question?

I understood everything before the last sentence but now I don't know how to approach this problem.

Any help is appreciated!