I am reading Nathan Jacobson's Basic Algebra I (2nd edition) in my private study. On p.227, there is a
THEOREM 4.4. Let $\eta:a\to\bar{a}$ be an isomorphism of a field $F$ onto a field $\bar{F}$, $f(x)\in F[x]$ be monic of positive degree, $\bar{f}(x)$ the corresponding polynomial in $\bar{F}[x]$ (under the isomorphism which extends $\eta$ and maps $x\to x$), and let $E$ and $\bar{E}$ be splitting fields of $f(x)$ and $\bar{f}(x)$ over $F$ and $\bar{F}$ respectively. Then $\eta$ can be extended to an isomorphism of $E$ onto $\bar{E}$. Moreover, the number of such extensions is $\le [E:F]$ and it is precisely $[E:F]$ if $\bar{f}(x)$ has distinct roots in $\bar{E}$.
Is there any example that strict inequality holds? I think that in such an example, $\bar{f}$ has to have an irreducible factor with multiple roots, meaning that $\bar{F}$ is an infinite field of finite characteristic. Jacobson has exhibited that such a field exists (e.g. the field of fractions of $(\mathbb Z/(p))[t]$ where $t$ is an indeterminate), but I am unable to come up with a concrete example of a strict inequality using this infinite field.
Best Answer
Let $K$ be a field of characteristic $p$ and $a\in K\setminus K^p$. Then it's well-known that $f=X^p-a$ is irreducible, and has only one root $b$ in the splitting field of $f$ (indeed, if $b$ is a root, then $f=(X-b)^p$).
Letting $E$ be that splitting field, clearly $E=K(b)$. Therefore, any automorphism of $E$ over $K$ is determined by where it sends $b$, but it must send $b$ to a root of $X^p-a$, therefore to itself. So $Aut(E/K)=\{id_E\}$, which has cardinal $1$, as opposed to $p=[E:K]$.
Taking $K=\mathbb{F}_p(t)$ and $a=t$ provides an example.