Algebraic integer

An algebraic number is an algebraic integer when it satisfies a monic polynomial with integer coefficients. Algebraic integers are closed under addition and multiplication.

The ring of integers ${\mathcal{O}}_K$ of a number field $K$ is the ring of all algebraic integers in $K$.

An order in a number field $K$ is a subring $A\subseteq {\mathcal{O}}_K$ containing $1$ and having full rank $[K:\mathbb{Q}]$ as an additive abelian group. Its finite index in the ring of integers of a number field is related to its discriminant by the discriminant-index formula for an integral lattice.

Every nonzero ideal of the ring of integers of a number field factors uniquely as a finite product of nonzero prime ideals. Thus each $x\in K^{\times }$ has integer valuations ${\mathrm{ord}}_{\mathfrak{p}}(x)$, almost all zero, determined by the factorization of its principal ideal.

If $\mathfrak{p}$ is a prime ideal of ${\mathcal{O}}_K$ above a rational prime $p$, its ramification index is the exponent $e_{\mathfrak{p}}$ of $\mathfrak{p}$ in the factorization of $(p)$. Equivalently, for the normalized ideal valuation, $e_{\mathfrak{p}}=v_{\mathfrak{p}}(p)$.

Let $B$ be the integral closure of a Dedekind domain $R$ in a finite separable extension $L/K$ of degree $n$. For a nonzero prime ideal $\mathfrak{p}$ of $R$, writeand let $f_{\mathfrak{q}/\mathfrak{p}}=[B/\mathfrak{q}:R/\mathfrak{p}]$. Then

For a prime number $p$ and $\alpha =\sqrt[p]{p}$, the polynomial $X^p-p$ is Eisenstein at $p$, andThe discriminant-index formula for an integral lattice reduces the proof to the prime $p$. At the unique prime ideal above $p$, one has $v(\alpha )=1$ and $v(p)=p$, which rules out an integral element $g(\alpha )/p$ with $g\notin p\mathbb{Z}[X]$.

For square-free $d\ne 0,1$, an integral basis is $\{1,(1+\sqrt{d})/2\}$ when $d\equiv 1(\mathrm{mod}4)$, and $\{1,\sqrt{d}\}$ otherwise. The discriminant is respectively $d$ and $4d$.

Since $2\not\equiv 1(\mathrm{mod}4)$, the ring of integers of $\mathbb{Q}(\sqrt{2})$ is $\mathbb{Z}[\sqrt{2}]$, with integral basis $1,\sqrt{2}$ and field discriminant eight.

The ring $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain for the absolute field norm. Approximate both rational coefficients of a quotient by nearest integers. If the errors are $u,w\in [-1/2,1/2]$, then $|u^2-2w^2|\le 1/2<1$, giving a remainder of strictly smaller absolute norm. In particular this ring is a principal ideal domain and has trivial ideal class group.