Read up on degrees of field extensions over other fields. The idea is, given an irreducible polynomial over a field, the degree of the field extension by a root of that polynomial is the degree of the minimal polynomial with that root. Just as you showed, has as a root. Now, can you prove that there is not a polynomial of smaller degree whose root is ? Easily. Assume there is a first degree polynomial over whose root is : . Now, plug in the root: , which is false. So, there is no degree one polynomial with as a root.
So, consider the field extension where , but is reducible over . For example, if , then . Now, consider the field extension over . It clearly has order 1 since any element of the field extended by is still in the field extended by .
So, chance are, you are probably overthinking it, because the book is just describing the expected behavior. It works just as you would expect it to. It turns out that these degrees are multiplicative. It is easy to check that has degree 4 over , and a minimal polynomial is .