Number field ($[K:\mathbb{Q}]<\infty$)

A number field is a finite field extension $K/\mathbb{Q}$ of the rational numbers. Its degree of a field extension $[K:\mathbb{Q}]$ is its dimension as a vector space over $\mathbb{Q}$. Every element is an algebraic number; the primitive element theorem also says that $K=\mathbb{Q}(\theta )$ for some single algebraic number $\theta$.

For example,The basis $1,\sqrt{2}$ makes arithmetic explicit: use $(\sqrt{2}{)}^2=2$ to reduce products. Every nonzero element has its inverse in the same field:The denominator cannot vanish unless $a=b=0$, since $\sqrt{2}$ is not a rational number. For instance, $(3+\sqrt{2}{)}^{-1}=(3-\sqrt{2})/7$ belongs to the number field, although it is not an algebraic integer.

A degree-$n$ number field has exactly $n$ distinct field embeddings into the complex numbers fixing $\mathbb{Q}$, because every finite field extension in characteristic zero is a separable field extension. For $\mathbb{Q}(\sqrt{2})$ these are the two real mapsBoth preserve addition and multiplication. Keeping both values, rather than just the positive choice of $\sqrt{2}$, is what makes the field norm and the Minkowski embedding of a number field useful.

A number field is totally imaginary if it has no embedding into the real numbers. Its embeddings into the complex numbers occur entirely in conjugate pairs. This is also called totally complex; it does not assert that every element of the field is nonreal.

A CM-field is a totally imaginary number field that is a quadratic extension of a totally real number field. Its nontrivial automorphism over that real subfield acts as complex conjugation in every complex embedding. Imaginary quadratic number fields are examples. An arbitrary totally imaginary number field need not have such a real subfield of index two.

For a CM-field, put $c$ for conjugation and $E_K={\mathcal{O}}_K^{\times }$. The map $u\mapsto u/c(u)$ takes values in the cyclic group ${\mu }_K$ of roots of unity: all conjugates have absolute value one, and the Kronecker theorem on algebraic integers in the unit disk applies. Its kernel is $E_{K^{+}}$, while its image on ${\mu }_K$ is ${\mu }_K^2$. It therefore embeds $E_K/(E_{K^{+}}{\mu }_K)$ into ${\mu }_K/{\mu }_K^2$, a group of order two. This proves the index bound without assuming that every unit is a real unit times a root of unity.

A number field is totally real when every embedding into the complex numbers has image in the real numbers. The maximal real subfield of a cyclotomic field is a basic example. Ordinary Hilbert class fields split at real places and therefore retain this property.

A pure cubic number field is a degree-three number field generated by a root of $X^3-m$ for a rational number $m$ that is not a rational cube. For a positive square-free integer $m>1$, this polynomial is an Eisenstein polynomial at any prime dividing $m$. The ring of integers of a number field depends on congruences at $3$, as expressed by the integral basis of a nonexceptional pure cubic field.

A quadratic number field is a number field of degree of a field extension two, necessarily of the form $\mathbb{Q}(\sqrt{d})$ for a square-free integer $d\ne 0,1$. Its two field embeddings send $a+b\sqrt{d}$ to $a\pm b\sqrt{d}$. If $d>0$, both images lie in the real numbers, giving a totally real number field; if $d<0$, they form a pair related by complex conjugation, giving an imaginary quadratic field. The ring of integers of a quadratic field depends on $d$ modulo four.^[ref]

The ring of integers is $\mathbb{Z}[\sqrt{10}]$, its class number is two, and its units are $\pm (3+\sqrt{10}{)}^n$ for $n\in \mathbb{Z}$. Factoring a norm-six principal ideal above 2 and 3 gives all solutionsRestriction to nonnegative $n$ gives only the positive-coordinate branch up to simultaneous sign. For example, $4-\sqrt{10}$ is a norm-six element requiring $n=-1$; adding conjugates repairs a nonnegative-index presentation.

An imaginary quadratic field is a quadratic number field with no real embedding. Its two complex embeddings are interchanged by complex conjugation. The endomorphism ring of an elliptic curve in characteristic zero, when larger than the integers, is an order in such a field; this is complex multiplication.

A fundamental discriminant is an integer that occurs as the field discriminant of a quadratic number field. For square-free $d$, it is $d$ when $d\equiv 1(\mathrm{mod}4)$ and $4d$ otherwise.