I would like an explanation for the fact stated in the title. To repeat:
Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of degree $n$?
Note that the statement might as well be false (though I would not bet on that). [EDIT: I thought I had a proof that the statement was true for $n=3$, but as Emil Jeřábek points out in his comments, this is false as well!]
This might be completely trivial, but I have no idea on how to address this question, and since it kinds of naturally arose in my research, I thought it would be appropriate to ask here.