Suppose $A$ is a ring containing a field $F$. Is ring homomorphism from $F$ to $A$ unique?

Question

It is derived from an exercise in Patrick Morandi's Field and Galois Theory, which states that if $a_1, \cdots, a_n \in A$ , there exists a unique ring homomorphism from $F[x_1, \cdots, x_n]$ to $A$ carrying each $x_i$ to $a_i$. I found this essentially equals to the question described in the title. How can one prove it? Thanks.

Answer 1

The answer to question in the title is No: there are three copies of $\mathbb Q(\sqrt[3]{2})$ inside $\mathbb C$.

The question in the title is not the same as the question in the body.

The answer to the question in the body is Yes, because $F[x_1, \dots, x_n]$ is a free $F$-algebra. And the exact result is that there is a unique $F$-algebra homomorphism $F[x_1, \dots, x_n] \to A$ with $x_i \mapsto a_i$, that is, a ring homomorphism that fixes $F$.