$F[a] \subseteq F(a)?$ I think this is probably an easy question, but I'd just like to check that I'm looking at it the right way.

Let $F$ be a field, and let $f(x) \in F[x]$ have a zero $a$ in some extension field $E$ of $F$. Define $F[a] = \left\{ f(a)\ |\ f(x) \in F[x] \right\}$. Then $F[a]\subseteq F(a)$.

The way I see this is that $F(a)$ contains all elements of the form $c_0 + c_1a + c_2a^2 + \cdots + c_na^n + \cdots$ ($c_i \in F$), hence it contains $F[a]$. Is that the "obvious" reason $F[a]$ is in $F(a)$?
And by the way, is $F[a]$ standard notation for the set just defined?

My preferred definition for $F[a]$ is the smallest ring that contains $F$ and $a$ and for $F(a)$ is the smallest field that contains $F$ and $a$. With these definitions, the inclusion is clear, because every field is a ring. The concrete descriptions in terms of polynomial and rational functions are then theorems of course.
