Lang's *Algebra*: definition of $F[\alpha]$ and why it's an integral domain?

Question

I am reading Lang's Algebra, namely the chapter about Fields.

1. The first thing which confused me is the following: how he defines $F[\alpha]$? Later he defines this as the smallest subfield of $E$ which contains both $F$ and $\alpha$. But in this context it is not clear what is $F[\alpha]$. Can anyone explain this, please?

2. Suppose we defined $F[\alpha]$ in some way. How it follows that this is integral domain (Lang calls it entire ring)?

Would be very grateful for help!

Answer 1

Lang makes the following definition in Chapter 2, $\S$1 (third paragraph, page 90, third edition):

Let $A$ be a subring of a ring $B$. Let $S$ be a subset of $B$ commuting with $A$; in other words, we have $sa = as$ for all $a \in A$ and $s \in S$. We denote by $A[S]$ the set of all elements $$ \sum a_{i_1,\dots,i_n} s_1^{i_1} \cdots s_n^{i_n}, $$ the sum ranging over a finite number of $n$-tuples $(i_1,\dots,i_n)$ of integers $\geq 0$, and $a_{i_1,\dots,i_n} \in A$, $s_1,\dots,s_n \in S$.

---

This gives you a definition of $F[\alpha]$. Take $A = F$, $B = E$ and $S = \{ \alpha \}$ to get that $F[\alpha]$ is the set of all elements $$ \sum_{i=0}^n a_i \alpha^i, $$ the sum ranging over all integers $n \geq 0$, and $a_i \in F$.

---

Why is $F[\alpha]$ an integral domain? This is because every element in $F[\alpha]$ is also an element of $E$: notice that $A[S] \subset B$ by the definition of $A[S]$.

Since $E$ is a field, it is an integral domain. Hence, any subring of $E$ is also an integral domain. In particular, $F[\alpha]$ is an integral domain.