In Artin there is a lemma that says if we have a field F and an extension F(a) (where a is a primitive element) and if b belongs to F(a) and b is a root of the minimal polynomial of a over F then F(a) = F(b).
I am not certain exactly why this is true. There is an F automorphism between the two and both have the same degree over F but I am not certain why that implies the above.
Thanks
If you have that $b\in F(a)$, and $a$ and $b$ are roots of the same irreducible polynomial over $F$, then $F\subset F(b) \subset F(a)$, and $[F(a):F]=[F(b):F]=$ the degree of the minimal polynomial. Hence $[F(a):F(b)]=1$, so they are equal.
In fact, you don't need $b$ to be root of the same irreducible polynomial: only of an irreducible polynomial of the same degree.