2026 07 17

I’ve been reading Algebraic Number Theory and Fermat’s Last Theorem by Ian Stewart and David Tall in a seminar this summer holiday.

I found that ring of algebraic integers has some very nice properties, which makes it behaves like the ring of rational integers $\mathbb{Z}$. For instance, we know:

The ring of integers $\mathfrak{O}$ of a number field $K$ has the following properties:

1. It is an ingegral domain, with the field of fractions $K$.
2. It is noetherian.
3. $\forall \alpha\in K$ satisfying a monic polynomial equation with coefficients in $\mathfrak{O}$, then $\alpha\in\mathfrak{O}$.
4. Every non-zero prime ideals of $\mathfrak{O}$ is maximal.

Isn’t it nice? Such kind of large famliy of rings, where all of which shares such well properties, has been really rare.