List of Commutative Fields

Introduction

This article lists every field that the corpus introduces — $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$, the finite fields, the $p$-adic fields, the rational function fields $k(x)$ and the Laurent series fields $k((t))$, the real algebraic numbers and the number fields — with the characteristic of each and the extensions of each. Every entry points to the article that introduces the object.

The list separates the fields of characteristic $0$, which contain a copy of $\mathbb{Q}$, from those of characteristic $p$, which contain a copy of $\mathbb{F}_p$; it separates the fields that are algebraically closed, such as $\mathbb{C}$, $\overline{\mathbb{Q}}$, $\overline{\mathbb{F}_p}$ and $\mathbb{C}_p$, from those that are only real closed, such as $\mathbb{R}$ and the real algebraic numbers; and it records the extensions and completions that the corpus builds from a given field.

The article introduces nothing and proves nothing. It records examples and non-examples side by side.

The Prime Fields and the Number Systems

Field	Characteristic	The prime field it contains	Extensions and closures	Introduced in
$\mathbb{Q}$	$0$	itself	the number fields; the completions $\mathbb{R}$ and $\mathbb{Q}_p$; the algebraic closure $\overline{\mathbb{Q}}$	The Rational Numbers
$\mathbb{R}$	$0$	$\mathbb{Q}$	$\mathbb{C} = \mathbb{R}(i)$, its algebraic closure; the real algebraic numbers as a proper subfield	The Real Numbers
$\mathbb{C}$	$0$	$\mathbb{Q}$	algebraically closed; already its own algebraic closure	The Complex Numbers
$\overline{\mathbb{Q}}$, the algebraic numbers	$0$	$\mathbb{Q}$	algebraically closed; the algebraic closure of $\mathbb{Q}$	Algebraically Closed Fields
$\overline{\mathbb{Q}} \cap \mathbb{R}$, the real algebraic numbers	$0$	$\mathbb{Q}$	the real closure of $\mathbb{Q}$; algebraic closure $\overline{\mathbb{Q}}$; countable and not order-complete	Real-Closed and Complete Ordered Fields
a number field $K$	$0$	$\mathbb{Q}$	finite over $\mathbb{Q}$; completions at its primes	Algebraic Number Theory
$\mathbb{Q}(\zeta_n)$, a cyclotomic field	$0$	$\mathbb{Q}$	degree $\varphi(n)$ over $\mathbb{Q}$; the cyclotomic subfields	Cyclotomic Fields
$\mathbb{Q}(\sqrt{d})$, a quadratic field	$0$	$\mathbb{Q}$	degree $2$ over $\mathbb{Q}$; imaginary for $d < 0$, real for $d > 0$	Algebraic Number Theory

A field of characteristic $0$ contains a copy of $\mathbb{Q}$, and the prime field is the intersection of all its subfields. The real algebraic numbers are the real closure of $\mathbb{Q}$ and are countable; the algebraic numbers $\overline{\mathbb{Q}}$ are its algebraic closure, so the algebraic closure of the real algebraic numbers is $\overline{\mathbb{Q}}$ rather than $\mathbb{C}$.

The Finite Fields and Characteristic $p$

Field	Characteristic	The prime field it contains	Extensions and subfields	Introduced in
$\mathbb{F}_p$	$p$	itself	$\mathbb{F}_q$ is a vector space over it; the prime subfield of every characteristic-$p$ field	Finite Fields
$\mathbb{F}_q$, $q = p^n$	$p$	$\mathbb{F}_p$	unique of order $q$; $\mathbb{F}_{p^m} \subseteq \mathbb{F}_{p^n}$ exactly when $m \mid n$	Finite Fields
$\overline{\mathbb{F}_p}$	$p$	$\mathbb{F}_p$	algebraically closed; the union of the $\mathbb{F}_{p^n}$; the residue field of $\mathbb{C}_p$	Finite Fields
any field of characteristic $p$	$p$	$\mathbb{F}_p$	contains $\mathbb{F}_p$ as its prime field; the Frobenius is injective	Fields

The finite fields are exhausted by this list, and for each prime power $q$ there is exactly one field of order $q$ up to isomorphism; the Frobenius map and the cyclic multiplicative group of order $q - 1$ are recorded in Finite Fields. The finite fields are fields of this list and not of the non-commutative list, by Wedderburn's little theorem.

The $p$-adic and Complete Valued Fields

Field	Characteristic	Valuation ring and residue field	Extensions and closures	Introduced in
$\mathbb{Q}_p$	$0$	$\mathbb{Z}_p$, maximal ideal $p\mathbb{Z}_p$, residue field $\mathbb{F}_p$, value group $\mathbb{Z}$	finite extensions; the algebraic closure $\overline{\mathbb{Q}_p}$; the completion $\mathbb{C}_p$	Absolute Values, Valuations and Completions
$\mathbb{C}_p$	$0$	$\overline{\mathbb{Z}_p}$, residue field $\overline{\mathbb{F}_p}$, value group $\mathbb{Q}$	algebraically closed and complete	Absolute Values, Valuations and Completions
$\overline{\mathbb{Q}_p}$	$0$	$\overline{\mathbb{Z}_p}$	not complete; its completion is $\mathbb{C}_p$	Absolute Values, Valuations and Completions
$\mathbb{R}$	$0$	the Archimedean completion of $\mathbb{Q}$	as in the first table	The Real Numbers

The two completions of $\mathbb{Q}$ at the Archimedean and the non-Archimedean places are $\mathbb{R}$ and $\mathbb{Q}_p$, by Ostrowski's theorem recorded in Absolute Values, Valuations and Completions. The completion preserves the value group and the residue field in the non-Archimedean case, which is why $\mathbb{Q}_p$ has value group $\mathbb{Z}$ and residue field $\mathbb{F}_p$.

Rational Function Fields and Laurent Series Fields

Field	Characteristic	Residue field and value group	Extensions and closures	Introduced in
$k(x)$, $k(x_1, \dots, x_n)$	$\operatorname{char} k$	—	the fraction field of $k[x]$; a purely transcendental extension	Fields, §19
$k((t))$	$\operatorname{char} k$	$k$, value group $\mathbb{Z}$	the completion of $k(t)$ at the $t$-adic valuation	Absolute Values, Valuations and Completions
$\mathbb{F}_p((t))$	$p$	$\mathbb{F}_p$, value group $\mathbb{Z}$	the completion of $\mathbb{F}_p(t)$	Absolute Values, Valuations and Completions
the Puiseux series field	$0$	—	the real closure of $\mathbb{R}(t)$ with $t$ infinite	Real-Closed and Complete Ordered Fields
$\operatorname{Frac}(R)$ of a domain $R$	$\operatorname{char} R$	—	the smallest field containing $R$	Localization and the Fraction Field

The rational function field $k(x)$ is the fraction field of the polynomial ring $k[x]$, and the Laurent series field $k((t))$ is its completion at the $t$-adic valuation. The two constructions differ by completion, and the residue field and the value group are the invariants that record the difference.

The Algebraic Closures of the Fields

Every field has an algebraic closure, unique up to isomorphism, and for the fields of this list the closure is known by name.

Field	Its algebraic closure	Introduced in
$\mathbb{Q}$	$\overline{\mathbb{Q}}$, the algebraic numbers	Algebraically Closed Fields
$\mathbb{R}$	$\mathbb{C} = \mathbb{R}(i)$	Galois Theory of ℂ/ℝ
the real algebraic numbers $\overline{\mathbb{Q}} \cap \mathbb{R}$	$\overline{\mathbb{Q}}$, not $\mathbb{C}$, since $\mathbb{C}$ is transcendental over $\mathbb{Q}$	Real-Closed and Complete Ordered Fields
$\mathbb{F}_p$, $\mathbb{F}_q$	$\overline{\mathbb{F}_p}$, the union of the $\mathbb{F}_{p^n}$	Finite Fields
$\mathbb{Q}_p$	$\overline{\mathbb{Q}_p}$, whose completion is $\mathbb{C}_p$	Absolute Values, Valuations and Completions
a number field $K$	$\overline{\mathbb{Q}}$	Algebraic Number Theory

The case of $\mathbb{R}$ is the Galois theory of $\mathbb{C}$ over $\mathbb{R}$: the extension has degree $2$, generated by a root of $x^2 + 1$, and $\mathbb{C}$ is algebraically closed by the fundamental theorem of algebra. The case of $\mathbb{F}_p$ is the union of the finite extensions $\mathbb{F}_{p^n}$, one for each $n$ inside the closure. For the real algebraic numbers the closure is not $\mathbb{C}$, because $\mathbb{C}$ contains transcendental elements over $\mathbb{Q}$.

Characteristic and Extensions at a Glance

Characteristic	The prime field	The fields of the corpus	Introduced in
$0$	$\mathbb{Q}$	$\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, $\overline{\mathbb{Q}}$, the real algebraic numbers, the number fields, $\mathbb{Q}_p$, $\mathbb{C}_p$, $k(x)$ and $k((t))$ for $k$ of characteristic $0$	The Rational Numbers, Finite Fields, Absolute Values, Valuations and Completions, Fields
$p$	$\mathbb{F}_p$	$\mathbb{F}_p$, $\mathbb{F}_q$, $\overline{\mathbb{F}_p}$, $\mathbb{F}_p((t))$, and $k(x)$, $k((t))$ for $k$ of characteristic $p$	Finite Fields, Absolute Values, Valuations and Completions, Fields

Warnings

Object	Why it is not a field	Introduced in
$\mathbb{H}$	a division ring, but not commutative; a skew field	Quaternion Algebra
the division ring of fractions of $A_1(k)$	not commutative	Ore Domains and Division Rings of Fractions
$\mathbb{Z}$, $\mathbb{Z}[x]$, $\mathbb{Z}[i]$	commutative domains that are not fields: not every nonzero element is a unit	The Integers, Examples of Rings and Fields
$\mathbb{O}$	a division algebra whose multiplication is not associative; not a ring	Octonion Algebra
the zero ring $\{0\}$	excluded by the convention $1 \neq 0$; a field has $1 \neq 0$	Rings, §2

A commutative division ring is a field, so $\mathbb{H}$ is excluded by commutativity alone and not by the absence of inverses. By Wedderburn's little theorem a finite division ring is a field, so there is no finite skew field and List of Division Rings and Skew Fields is left with the infinite ones.

Summary

This article has listed the fields of the corpus with their characteristic and their extensions. The characteristic-$0$ fields are the number systems and their closures and completions; the characteristic-$p$ fields are the finite fields, their algebraic closure and the Laurent series fields over a field of characteristic $p$; the function fields $k(x)$ and the Laurent series fields $k((t))$ carry the characteristic of their ground field; and the real algebraic numbers, the real closure of $\mathbb{Q}$, are the field that is real closed without being order-complete. The extensions of each are recorded beside it: the algebraic closures $\overline{\mathbb{Q}}$, $\overline{\mathbb{F}_p}$ and $\mathbb{C}_p$, the completion $\mathbb{C}_p$ of $\overline{\mathbb{Q}_p}$, and the algebraic closure $\mathbb{C} = \mathbb{R}(i)$ of $\mathbb{R}$.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.

Symbol	Meaning
$\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$	The rational, real and complex fields
$\overline{\mathbb{Q}}$, $\overline{\mathbb{F}_p}$, $\overline{\mathbb{Q}_p}$	Algebraic closures
$\mathbb{F}_p$, $\mathbb{F}_q$	The finite fields of order $p$ and $q = p^n$
$\mathbb{Q}_p$, $\mathbb{Z}_p$, $\mathbb{C}_p$	$p$-adic numbers, $p$-adic integers, complete algebraic closure
$k(x)$, $k((t))$, $\mathbb{F}_p((t))$	Rational function field, Laurent series fields
$\mathbb{Q}(\zeta_n)$, $\mathbb{Q}(\sqrt{d})$, $K$	Cyclotomic, quadratic and number fields
$\overline{\mathbb{Q}} \cap \mathbb{R}$	The real algebraic numbers
$\operatorname{Frac}(R)$	Field of fractions
$\varphi(n)$	Euler totient, the degree $[\mathbb{Q}(\zeta_n):\mathbb{Q}]$
$\operatorname{char} k$	The characteristic of a field
$\mathbb{H}$	The quaternions; a division ring, not a field