§ Integral elements of a ring form a ring [TODO ]
- An integral element of a field (imagine ) relative to an integral domain (imagine ) is the root of a monic polynomial in .
- So for example, in the case of over , the element is integral as it is a root of .
- On the other hand, the element is not integral. Intuitively, if we had a polynomial of which it is a root, such a polynomial would be divisible by (which is the minimal polynomial for ). But is not monic.
- Key idea: take two element which are roots of polynomial .
- Create the polynomial (for construction) given by . See that has both and as roots, and lies in .