The following is with regards to Lemma 3.18 from Field and Galois Theory by Patrick Morandi.
Let $\sigma : F \to F'$ be a field isomorphism, $K$ be a field extension of F, and $K'$ be a field extension of $F'$. Suppose that $K$ is a splitting field of $\{f_i\}$ over F and that $\tau: K \to K'$ is a homomorphism with $\tau|_F = \sigma$. If $f_i':=\sigma(f_i)$, then $\tau(K)$ is a splitting field of $\{f_i'\}$ over $F'$.
To clarify, since $\sigma$ and $\tau$ are field homomorphisms, there are natural induced ring homomorphisms $\sigma':F[x]\to F'[x]$ and $\tau':K[x]\to K'[x]$. So when he writes $\sigma(f)$ or $\tau(f)$, he really means $\sigma'(f)$ or $\tau'(f)$.
I have two questions regarding the proof. The first is that Morandi assumes that $\tau(K)$ is a field. Now if we assume that ring homomorphisms must map $1_F\to 1_{F'}$, then it will be the case $\tau$ is injective and thus, $\tau(K)$ is a field. Should this be the case?
For my second question, let $X$ be the set of all the roots of all the $f_i$ in $K$. Then $K=F(X)$. Morandi claims that $$\tau(F(X))=F'(\tau(X)).$$ I was able to show that $F'(\tau(X))\subseteq \tau(F(X))$, but I am not convinced of the other inclusion. Could someone kindly explain this?
To see why $\tau(K)$ is a field, you are correct in saying that $\tau$ is injective. Furthermore, it is surjective. Hence, they are isomorphic as rings so in particular it has a field structure. To see the other inclusion, note that elements in $\tau(F(X))$ is simply $\tau$ of polynomials with coefficients $F$ over the elements of $X.$ This is because the extension is algebraic. Take any monomial of the form $f \alpha_1^{e_1} \ldots \alpha_k^{e_k}$ with $f \in F$ and $\alpha_i \in X.$ Then $\tau(f\alpha_1^{e_1} \ldots \alpha_k^{e_k})$ is just $\tau(f)\tau(\alpha_1)^{e_1}\ldots \tau(\alpha_k)^{e_k}$ which is an element in $F'(\tau(X)).$