I'm reading A Book Of Abstract Algebra by Charles C. Pinter. On page 314 is the following theorem:
Let $h:F_1\to F_2$ be an isomorphism, and let $p(x)$ be irredicible in $F_1[x]$. Suppose $a$ is a root of $p(x)$, and $b$ is a root of $h(p(x))$. Then $h$ can be extended to an isomorphism $\bar h:F_1(a)\to F_2(b)$, and $\bar h(a)=b$.
By $h(p(x))$ the author is referring to the obvious extension of $h$ to $F_1[x]$ (applying $h$ to each coefficient).
Noting that each element of $F_1(a)$ can be expressed in a unique fashion as $\sum c_i a^i$, the author defines $$\bar h(\sum c_i a^i)=\sum h(c_i) b^i$$
This seems to me to be well-defined, always a homomorphism, and bijective as long as $a$ has the same degree over $F_1$ as $b$ has over $F_2$. The author's requirement that $b$ be a root of $h(p(x))$ seems unnecessarily strong.
Question: Is this theorem true so long as $a$ has the same degree over $F_1$ as $b$ has over $F_2$?