A Universe of Sorts

An integral element of a field $L$ (imagine $\mathbb C$ ) relative to an integral domain $A$ (imagine $\mathbb Z$ ) is the root of a monic polynomial in $A$ .

So for example, in the case of $\mathbb C$ over $\mathbb Z$ , the element $i$ is integral as it is a root of $p(x) = x^2 + 1$ .
On the other hand, the element $1/2$ is not integral. Intuitively, if we had a polynomial of which it is a root, such a polynomial would be divisible by $2x - 1$ (which is the minimal polynomial for $1/2$ ). But $2x - 1$ is not monic.
Key idea: take two element $a, b$ which are roots of polynomial $p(x), q(x) \in A[x]$ .
Create the polynomial $c(x)$ (for construction) given by $c(x) \equiv p(x)q(x) \in A[x]$ . See that $c(x)$ has both $a$ and $b$ as roots, and lies in $A[x]$ .

§ Integral Elements of a Ring Form a Ring [TODO ]