Explanation about a theorem from Basic Algebra I, Jacobson I have a doubt about theorem 4.4, page 227 of Basic Algebra I.
Specifically, I don't understand why it says that the number of extensions is $\le [E:F]$, unless $\bar f (x)$ has distinct roots in $\bar E $; in the demonstration Jacobson uses the lemma below, that is true because it refers to generical extension fields. For example if I take the roots of $x^3-2\in \mathbb Q [x]$, let's call them $a, b$ and $c $, then $[\mathbb Q(a):\mathbb Q]=3$ but I can have only two extension monomorphisms from $\mathbb Q(a)$ to $\mathbb Q(a,b) $. However in the theorem 4.4 we are talking about splitting fields, so the fact that the number of extensions is less than $[E:F] $ implies thae number of distinct roots of $f (x) $ in $E$ is different from the number of distinct roots of $\bar f (x) $ in $\bar E$. How is it possible? (Can you make me an example); and in that case, an isomorphism between $E $ and $\bar E$ must send two distinct roots of $f $ in a single element of $\bar E$ right? (Assuming that $f $ has more distinct roots in $E $ than $\bar f (x) $ in $\bar E$). I'm not very sure of these conclusions, so if anyone can give me a clarification I'll be grateful.
 A: It seems that you are misinterpreting the theorem.
Jacobson Basic Algebra I, Theorem 4.4: Suppose we have a field isomorphism $\eta\colon F\to \bar F$, a polynomial $f\in F[x]$ and its image $\bar f$ under $\eta$, and their respective splitting fields $E$ and $\bar E$. Then
(1) we can extend $\eta$ to an isomorphism $\eta'\colon E\to \bar E$.
(2) the number of possible extensions of $\eta$ is at most $[E:F]$.
(3) the number of possible extensions of $\eta$ is precisely $[E:F]$ whenever $\bar f$ has distinct roots in $\bar E$.
As you rightly note, since any extension $\eta'$ is an isomorphism, the number of distinct roots of $f$ in $E$ equals the number of distinct roots of $\bar f$ in $\bar E$. I agree that it would make more sense to replace (3) by the equivalent condition that $f$ has distinct roots in $E$.
It seems that the theorem is written in this way because of the first lemma used in the proof, which talks about extending $\eta$ to a field homomorphism $F(\alpha)$ to some  field extension $\bar E$ of $\bar F$. Such an extension exists if and only if $\bar g$ has a root in $\bar E$, where $g$ is the minimal polynomial of $\alpha$ over $F$.
