Suppose that F is a field. Prove that F[x, y, z] is an integral domain.
As a starting point, I would like to use the theorem that states that every field is an integral domain and then work from there.
However, the textbook I am using (Abstract Algebra theory and applications) has this theorem instead: "Theorem 16.16. Every finite integral domain is a field". I thought this was incorrect, but since the textbook definitely does not have any errors, could someone explain the textbook's theorem for me? Does this have to do with finite fields only? Thank you
If $R$ is a finite integral domain, consider any nonzero $a \in R$ and define $\varphi : R \to R$ by $\varphi(x) = ax$. Because $R$ is an integral domain, the cancellation law holds, thus $\varphi$ is an injective map. Because $R$ is finite, $\varphi$ must also the be surjective. Thus there must exist some $b \in R$ such that $ab = 1$, i.e. $a$ is a unit. We conclude that every nonzero element of $R$ is a unit, hence $R$ is a field.
For your other question, more generally it is true that if $R$ is an integral domain, then so is $R[x]$. To see this, consider two nonzero elements $f(x),g(x) \in R[x]$ and let $a_n,b_m$ be their respective leading terms. Since $f$ and $g$ are nonzero, so are $a_n$ and $b_m$, hence $a_nb_m$ is nonzero, thus $f(x)g(x) \neq 0$. Your result then follows by induction since $R[x_1,\ldots,x_n] = R[x_1,\ldots,x_{n-1}][x_n]$.