The Rational Numbers ($\mathbb{Q}$)

Introduction

The rational numbers are the field of fractions of the integers: the smallest field containing $\mathbb{Z}$, equivalently the smallest field of characteristic $0$, equivalently the prime field of every field of characteristic $0$. Everything about $\mathbb{Q}$ that is algebraic follows from this one construction: the field axioms, the absence of proper subfields, the uniqueness of the ordering, the triviality of the automorphism group, and the fact that every element is a ratio of two integers in essentially one way. What $\mathbb{Q}$ lacks is completeness, and the two ways of repairing that defect, by order and by absolute value, produce $\mathbb{R}$ and the fields $\mathbb{Q}_p$.

This article carries out the construction of $\mathbb{Q}$ from $\mathbb{Z}$, establishes its universal property, records its order and divisibility structure, and describes its subrings, its unit group, its orderings and its completions. The analysis of $\mathbb{Q}$ as a metric space belongs to the companion treatments of the real numbers and of valuations; the completions are named here and constructed in Absolute Values, Valuations and Completions and The Real Numbers.

Throughout, $\mathbb{Z}$ is the ring of integers and $\mathbb{Q} = \operatorname{Frac}(\mathbb{Z})$. Divisibility, prime factorization and localization are from Integral Domains, Unique Factorisation Domains and Localization and the Fraction Field.

The Field of Fractions of $\mathbb{Z}$

Construction

Theorem. $\mathbb{Z}$ is an integral domain, and its field of fractions

$$ \mathbb{Q} = \operatorname{Frac}(\mathbb{Z}) = (\mathbb{Z} \times (\mathbb{Z} \setminus \{0\}))/\sim, \qquad (n,m) \sim (k,\ell) \iff n\ell = mk, $$

with addition and multiplication defined by

$$ \frac{n}{m} + \frac{k}{\ell} = \frac{n\ell + mk}{m\ell}, \qquad \frac{n}{m} \cdot \frac{k}{\ell} = \frac{nk}{m\ell}, $$

is a field of characteristic $0$.

Proof. $\mathbb{Z}$ is an integral domain, so the relation $\sim$ is an equivalence relation and the displayed operations are well defined; the general construction of the fraction field is given in Localization and the Fraction Field. In $\mathbb{Q}$ the class of $\tfrac{n}{1}$ is nonzero whenever $n \neq 0$, and $n \cdot 1 = \tfrac{n}{1} \neq 0$ for every $n \geq 1$, so the characteristic is $0$.

Theorem (universal property). Let $K$ be a field and let $\varphi : \mathbb{Z} \to K$ be an injective ring homomorphism. Then there is a unique ring homomorphism $\Phi : \mathbb{Q} \to K$ with $\Phi(\tfrac{n}{1}) = \varphi(n)$; it is injective, and its image is the prime field of $K$.

Proof. The map $\Phi(\tfrac{n}{m}) = \varphi(n)\varphi(m)^{-1}$ is well defined because $\varphi(m) \neq 0$ for $m \neq 0$, and it is a ring homomorphism; uniqueness is forced because every rational is a quotient of images of integers. Injectivity: a field homomorphism is injective.

Corollary. $\mathbb{Q}$ is the prime field of characteristic $0$: it is the smallest subfield of any field of characteristic $0$, and every field of characteristic $0$ contains a copy of $\mathbb{Q}$ as its prime field. Up to isomorphism, $\mathbb{Q}$ is the only field of characteristic $0$ whose only subfield is itself, and it is the initial object in the category of fields of characteristic $0$.

Basic Arithmetic

Theorem (fundamental theorem of arithmetic). Every nonzero integer has a factorization into primes, unique up to order and signs, and consequently every $a \in \mathbb{Q}^\times$ has a unique expression

$$ a = \operatorname{sgn}(a) \prod_p p^{v_p(a)}, \qquad v_p(a) \in \mathbb{Z},\ \text{finitely many nonzero}, $$

where $\operatorname{sgn}(a) = \pm 1$ and $v_p$ is the $p$-adic valuation of Absolute Values, Valuations and Completions.

Proof. The factorization of integers is the unique factorization property of $\mathbb{Z}$ as a Euclidean domain, from Euclidean Domains; the rational statement follows by writing $a = n/m$ and subtracting the exponent vectors of $m$ from those of $n$, the uniqueness following from uniqueness for $n$ and $m$.

Corollary (the unit group). There is a group isomorphism

$$ \mathbb{Q}^\times \cong \mathbb{Z}/2\mathbb{Z} \oplus \bigoplus_p \mathbb{Z}, $$

the sign giving the $\mathbb{Z}/2$ and the exponent of each prime giving a copy of $\mathbb{Z}$.

Corollary. $\mathbb{Q}$ is a field; hence its only ideals are $0$ and $\mathbb{Q}$, it is a principal ideal domain and a unique factorization domain in the trivial sense, and so is $\mathbb{Q}[x]$, which is moreover a Euclidean domain for the degree function.

The Order of $\mathbb{Q}$

The Ordering

Definition. A rational $a = \tfrac{n}{m}$ is positive if $nm > 0$ in $\mathbb{Z}$; the order is $a < b$ iff $b - a$ is positive.

Theorem. The relation so defined is a total order on $\mathbb{Q}$ making it an ordered field, and it is the only ordering of $\mathbb{Q}$.

Proof. Well-definedness: if $\tfrac{n}{m} = \tfrac{k}{\ell}$ then $n\ell = mk$ and $nm$ has the same sign as $k\ell$, since $nm\ell^2 = m^2k\ell$ and $\ell^2, m^2 > 0$. The order axioms reduce to the corresponding facts in $\mathbb{Z}$, and multiplication by a positive rational preserves positivity by the sign rules. Uniqueness: in any ordering of a field, $1 > 0$ by the basic rules of Ordered Fields, hence $n \cdot 1 > 0$ for $n \geq 1$ and $n \cdot 1 < 0$ for $n \leq -1$; so the sign of every integer is forced, and then the sign of every ratio is forced, giving the positive cone displayed.

Corollary. $\mathbb{Q}$ is an Archimedean ordered field and the smallest one: it embeds as an ordered subfield in every ordered field. Its automorphism group is trivial, $\operatorname{Aut}(\mathbb{Q}) = 1$, and the only order-preserving automorphism of $\mathbb{Q}$ is the identity.

Theorem (the embedding of $\mathbb{Z}$ and density). The map $\mathbb{Z} \to \mathbb{Q}$, $n \mapsto \tfrac{n}{1}$ is an injective order-preserving ring homomorphism, and $\mathbb{Z}$ is unbounded above in $\mathbb{Q}$. The order on $\mathbb{Q}$ is dense: for $a < b$ there is $c$ with $a < c < b$, for instance $c = \tfrac{a+b}{2}$.

Proof. Injectivity and order-preservation are immediate from the construction. Unboundedness: for any $\tfrac{n}{m}$ with $m > 0$ the integer $\lvert n \rvert + 1$ exceeds it, since $\tfrac{n}{m} \leq n < n+1$ for $n \geq 0$ and $\tfrac{n}{m} < 0 < \lvert n \rvert + 1$ for $n < 0$. Density: $2 = 1 + 1 > 0$, so $2^{-1} > 0$ and $a = \tfrac{a+a}{2} < \tfrac{a+b}{2} < \tfrac{b+b}{2} = b$.

Remark. Density is a property of the order alone and does not make $\mathbb{Q}$ complete: the set $\{a \in \mathbb{Q} : a > 0,\ a^2 < 2\}$ is nonempty and bounded above and has no least upper bound in $\mathbb{Q}$, by the classical irrationality of $\sqrt2$. Completeness is treated in Real-Closed and Complete Ordered Fields and The Real Numbers.

Subrings, Ideals and Localizations

The Subrings of $\mathbb{Q}$

Theorem. Every subring $R \subseteq \mathbb{Q}$ is a localization of $\mathbb{Z}$: there is a set of primes $S$ such that

$$ R = \mathbb{Z}_S = \left\{\frac{n}{m} \in \mathbb{Q} : n \in \mathbb{Z},\ m \in \mathbb{Z}\setminus\{0\},\ \text{every prime dividing } m \text{ lies in } S\right\}. $$

Conversely each such $\mathbb{Z}_S$ is a subring of $\mathbb{Q}$ containing $\mathbb{Z}$, and the correspondence $S \mapsto \mathbb{Z}_S$ is bijective between sets of primes and subrings of $\mathbb{Q}$ containing $\mathbb{Z}$.

Proof sketch. Given $R$, let $S = \{p \text{ prime} : \tfrac1p \in R\}$. If $a = \tfrac{n}{m} \in R$ in lowest terms, then $\gcd(n,m) = 1$, so there are integers $u, v$ with $un + vm = 1$, whence $\tfrac{1}{m} = u\tfrac{n}{m} + v \in R$ by the Bézout relation; multiplying $\tfrac1m$ by $m/p$ for a prime $p \mid m$ gives $\tfrac1p \in R$, so every prime divisor of $m$ lies in $S$ and $R \subseteq \mathbb{Z}_S$. Conversely, if $a = \tfrac{n}{m}$ with every prime divisor of $m$ in $S$, then $\tfrac1m$ is a product of the elements $\tfrac1p \in R$ and their powers, so $\tfrac1m \in R$ and $a = n \cdot \tfrac1m \in R$; hence $\mathbb{Z}_S \subseteq R$. Bijectivity: $S$ is recovered from $\mathbb{Z}_S$ as the primes with $\tfrac1p \in \mathbb{Z}_S$.

Corollary. $\mathbb{Z}$ and $\mathbb{Q}$ are the extreme cases $S = \emptyset$ and $S = $ all primes; the intermediate rings $\mathbb{Z}_S$ are precisely the rings between $\mathbb{Z}$ and $\mathbb{Q}$, and each is a principal ideal domain obtained from $\mathbb{Z}$ by inverting the primes in $S$. The rings $\mathbb{Z}_{(p)}$ for a single prime $p$, the localizations of Localization and the Fraction Field, are the valuation rings of the $p$-adic valuations.

Ideals and Modules over $\mathbb{Q}$

Proposition. The only ideals of $\mathbb{Q}$ are $0$ and $\mathbb{Q}$; consequently $\mathbb{Q}$ is a field, every nonzero element is a unit, and $\mathbb{Q}$ has no proper nonzero quotients.

Proof. If $I \neq 0$ contains $a \neq 0$ then $1 = a \cdot a^{-1} \in I$, so $I = \mathbb{Q}$.

Proposition. $\mathbb{Z}$-submodules of $\mathbb{Q}$ need not be ideals, and are not classified by a single invariant: the additive subgroups of $\mathbb{Q}$ include $\mathbb{Z}$, $\mathbb{Z}[\tfrac12]$, $\mathbb{Z}_{(p)}$ and the $p$-primary subgroups, and the classification of the subgroups of $\mathbb{Q}$ belongs to the theory of abelian groups. The finitely generated $\mathbb{Z}$-submodules of $\mathbb{Q}$ are exactly the cyclic ones, $\mathbb{Z}\tfrac{n}{m}$, and the structure theory of modules over $\mathbb{Z}$ belongs to Modules.

$\mathbb{Q}$ as a Field: Algebraic Properties

Theorem. The following hold for $\mathbb{Q}$.

(a) $\mathbb{Q}$ has characteristic $0$, and every field of characteristic $0$ contains a copy of $\mathbb{Q}$.

(b) $\mathbb{Q}$ is countable.

(c) $\mathbb{Q}$ is not algebraically closed: the polynomial $x^2 - 2$ has no rational root.

(d) The algebraic closure of $\mathbb{Q}$ is the field $\overline{\mathbb{Q}}$ of algebraic numbers, which is countable; $\mathbb{Q}$ has extensions of every finite degree, and its absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is a large profinite group.

(e) $\mathbb{Q}$ is a perfect field, and every finite extension of $\mathbb{Q}$ is separable.

(f) $\mathbb{Q}$ has a unique ordering, and it is Archimedean; it is not order-complete and it is not real closed.

(g) $\operatorname{tr.deg}_{\mathbb{Q}} \mathbb{Q} = 0$, and every extension of $\mathbb{Q}$ that is algebraic over $\mathbb{Q}$ has transcendence degree $0$ over $\mathbb{Q}$.

Proof. (a) is the universal property. (b) $\mathbb{Q}$ is the image of the countable set $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$. (c) $\sqrt2 \notin \mathbb{Q}$ by the classical parity argument. (d) The algebraic closure is constructed in Algebraically Closed Fields; $x^n - 2$ is irreducible over $\mathbb{Q}$ for every $n$ by Eisenstein, giving extensions of every degree. (e) A field of characteristic $0$ is perfect. (f) The ordering is unique by the theorem above; $\mathbb{Q}$ is Archimedean by Ordered Fields; incompleteness is the remark above. (g) An algebraic extension has transcendence degree $0$ over its base field, and $\mathbb{Q}$ is algebraic over itself.

Remark (the place of $\mathbb{Q}$ among fields). $\mathbb{Q}$ is the prime field of characteristic $0$ and the initial object among such fields; it is not algebraically closed, not real closed, and not order-complete. Its completions are $\mathbb{R}$ and the fields $\mathbb{Q}_p$, which are locally compact, while $\mathbb{Q}$ itself is not locally compact in the usual topology. The rationals are the field in which the arithmetic of $\mathbb{Z}$ becomes invertibility of the primes, and the two families of completions, together with the product formula, are the starting point of algebraic number theory.

Extensions of $\mathbb{Q}$

Finite Extensions

Theorem. Every finite extension of $\mathbb{Q}$ is separable and simple: it is of the form $\mathbb{Q}(\alpha)$ for some algebraic $\alpha$, and its degree is the degree of the minimal polynomial of $\alpha$.

Proof. Every field of characteristic $0$ is perfect, so every algebraic extension is separable, by Splitting Fields and Algebraic Closure; in characteristic $0$ every finite extension is simple by the theorem of the primitive element, since a finite separable extension is simple (Field Extensions). The degree statement is the definition of the minimal polynomial, from Field Extensions.

Theorem. The algebraic closure $\overline{\mathbb{Q}}$ is countable, and $[\overline{\mathbb{Q}}:\mathbb{Q}]$ is infinite: there are extensions of $\mathbb{Q}$ of every finite degree.

Proof. The polynomial $x^n - 2$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion at $2$, so $\mathbb{Q}(\sqrt[n]{2})$ has degree $n$; hence the degree of $\overline{\mathbb{Q}}$ exceeds every integer. For countability, $\overline{\mathbb{Q}}$ is the union of the splitting fields of countably many polynomials over $\mathbb{Q}$, each finite and hence countable, so the union is countable.

Quadratic Fields

Theorem. The quadratic extensions of $\mathbb{Q}$, up to isomorphism, are the fields $\mathbb{Q}(\sqrt{d})$ for squarefree integers $d \neq 0, 1$, and

$$ \mathbb{Q}(\sqrt{d}) \cong \mathbb{Q}(\sqrt{d'}) \iff d/d' \in (\mathbb{Q}^\times)^2 . $$

The square classes $\mathbb{Q}^\times/(\mathbb{Q}^\times)^2$ form a vector space over $\mathbb{F}_2$ with basis the class of $-1$ and the classes of the primes, so there are countably many quadratic extensions of $\mathbb{Q}$.

Proof. A quadratic extension is generated by an element $\alpha$ with $\alpha^2 = q \in \mathbb{Q}$, and writing $q$ with squarefree numerator and denominator gives $\mathbb{Q}(\alpha) = \mathbb{Q}(\sqrt d)$ for a squarefree integer $d$. The displayed equivalence follows because an isomorphism of quadratic fields must send $\sqrt d$ to an element whose square is $d$. The description of the square classes is the unit group computation $\mathbb{Q}^\times \cong \mathbb{Z}/2\mathbb{Z} \oplus \bigoplus_p \mathbb{Z}$ read modulo squares: the sign class gives $-1$ and each free generator gives the class of a prime.

Cyclotomic Fields

Theorem. Let $\zeta_n$ be a primitive $n$-th root of unity. Then $\mathbb{Q}(\zeta_n)$ is the splitting field of $x^n - 1$ over $\mathbb{Q}$, it has degree $\varphi(n)$ over $\mathbb{Q}$, and

$$ \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \cong (\mathbb{Z}/n\mathbb{Z})^\times , $$

so $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is an abelian Galois extension. The cyclotomic fields, as $n$ varies, generate the maximal abelian extension of $\mathbb{Q}$: every finite abelian extension of $\mathbb{Q}$ is contained in some $\mathbb{Q}(\zeta_n)$, by the Kronecker–Weber theorem.

Proof. The roots of $x^n - 1$ are the powers of $\zeta_n$, so the field generated by $\zeta_n$ is the splitting field; an automorphism over $\mathbb{Q}$ is determined by $\zeta_n \mapsto \zeta_n^m$ with $\gcd(m,n) = 1$, giving the isomorphism onto $(\mathbb{Z}/n\mathbb{Z})^\times$, of order $\varphi(n)$; that the degree is exactly $\varphi(n)$ is the irreducibility of the cyclotomic polynomial, proved for prime $n$ by Eisenstein's criterion in Unique Factorisation Domains, and standard in general. Kronecker–Weber is a standard theorem of algebraic number theory, stated here for the structure of $\overline{\mathbb{Q}}$.

Example (the splitting field of $x^3 - 2$). The polynomial $x^3 - 2$ is irreducible over $\mathbb{Q}$, so $\mathbb{Q}(\sqrt[3]{2})$ has degree $3$; it is not a normal extension, because the other two roots are $\omega \sqrt[3]{2}$ and $\omega^2 \sqrt[3]{2}$ with $\omega = \zeta_3$ a primitive cube root of unity. The normal closure is $\mathbb{Q}(\sqrt[3]{2}, \omega)$, of degree $6$ over $\mathbb{Q}$, with Galois group $S_3$ acting on the three roots, as computed in Galois Theory.

Endomorphisms and Derivations of $\mathbb{Q}$

Theorem. $\operatorname{End}(\mathbb{Q}) = \{\mathrm{id}\}$ and $\operatorname{Der}(\mathbb{Q}) = 0$: every unital ring endomorphism of $\mathbb{Q}$ is the identity, every derivation of $\mathbb{Q}$ into itself is zero, and the only non-unital ring homomorphisms $\mathbb{Q} \to \mathbb{Q}$ are the identity and the zero map.

Proof. A unital ring homomorphism $\varphi : \mathbb{Q} \to \mathbb{Q}$ has kernel an ideal, so either $0$ or $\mathbb{Q}$; the kernel is not $\mathbb{Q}$ because $\varphi(1) = 1 \neq 0$, so $\varphi$ is injective, whence $\varphi(n) = n$ for integers and $\varphi(n/m) = \varphi(n)\varphi(m)^{-1} = n/m$. Dropping the unital condition adds the zero map and nothing else, since every other homomorphism carries $1$ to an idempotent of $\mathbb{Q}$ and the only nonzero idempotent is $1$. For derivations, $D(1) = D(1 \cdot 1) = D(1) + D(1)$ gives $D(1) = 0$, so $D(n) = 0$ for all integers and $0 = D(1) = D(m \cdot \tfrac1m) = m D(\tfrac1m)$, whence $D(\tfrac1m) = 0$ and $D(\tfrac{n}{m}) = 0$.

Corollary. $\operatorname{Aut}(\mathbb{Q}) = 1$, in agreement with the rigidity of the prime field observed in Ring and Field Automorphisms; and $\mathbb{Q}$ carries no nonzero derivation, so the tangent-space intuitions of differential algebra are vacuous over the rationals.

Completions and the Places of $\mathbb{Q}$

Ostrowski and the Places

Theorem (Ostrowski, restated from Absolute Values, Valuations and Completions). Every nontrivial absolute value on $\mathbb{Q}$ is equivalent to the usual absolute value $\lvert \cdot \rvert_\infty$ or to one of the $p$-adic absolute values $\lvert \cdot \rvert_p$. Hence the places of $\mathbb{Q}$ are $\infty$ and the primes $p$.

Corollary (the completions of $\mathbb{Q}$). The nontrivial completions of $\mathbb{Q}$ are

$$ \widehat{\mathbb{Q}}_\infty = \mathbb{R}, \qquad \widehat{\mathbb{Q}}_p = \mathbb{Q}_p \ (p \text{ prime}), $$

and no two of these are isomorphic: $\mathbb{R}$ is Archimedean and orderable, while each $\mathbb{Q}_p$ is not orderable, and $\mathbb{Q}_p \not\cong \mathbb{Q}_q$ for $p \neq q$ because the torsion subgroup of $\mathbb{Q}_p^\times$ is $\mu_{p-1}$, so two of these fields isomorphic as fields would force $p - 1 = q - 1$.

Theorem (product formula). For every $a \in \mathbb{Q}^\times$,

$$ \lvert a \rvert_\infty \prod_p \lvert a \rvert_p = 1 . $$

Proof. Immediate from the prime factorization $a = \operatorname{sgn}(a)\prod_p p^{v_p(a)}$, as in Absolute Values, Valuations and Completions.

The Local Structure

Proposition. For each prime $p$, the valuation ring of $\lvert \cdot \rvert_p$ on $\mathbb{Q}$ is the localization

$$ \mathbb{Z}_{(p)} = \left\{\frac{n}{m} : p \nmid m\right\} = \mathbb{Z}_S \ \text{with} \ S = \{\text{primes} \neq p\}, $$

with maximal ideal $p\mathbb{Z}_{(p)}$ and residue field $\mathbb{F}_p$; its completion is $\mathbb{Z}_p$. The field $\mathbb{Q}$ is recovered from each localization by inverting the remaining primes, and $\mathbb{Q}$ is the localization of $\mathbb{Z}$ at the multiplicative set of all nonzero integers.

Proof. The identification of the valuation ring is the previous section; the residue field and maximal ideal are computed from the valuation, and the completion statement is from Absolute Values, Valuations and Completions.

Summary

$\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$ and the prime field of characteristic $0$: it embeds as the smallest subfield in every field of characteristic $0$, its construction is the localization of $\mathbb{Z}$ at the nonzero integers, and it is characterised up to isomorphism as the unique field of characteristic $0$ with no proper subfield. It is countable, perfect, of characteristic $0$, and its only ideals are $0$ and $\mathbb{Q}$; its unit group is $\mathbb{Q}^\times \cong \mathbb{Z}/2\mathbb{Z} \oplus \bigoplus_p \mathbb{Z}$, the free abelian group on the primes modulo sign, by the fundamental theorem of arithmetic.

The ordering of $\mathbb{Q}$ is unique, Archimedean and dense, and makes $\mathbb{Q}$ the smallest ordered field; the order is not complete, and $\mathbb{Q}$ is neither real closed nor algebraically closed, with algebraic closure $\overline{\mathbb{Q}}$. Every subring of $\mathbb{Q}$ is a localization $\mathbb{Z}_S$ for a set of primes $S$, and the rings between $\mathbb{Z}$ and $\mathbb{Q}$ are exactly these localizations, with the $p$-localizations $\mathbb{Z}_{(p)}$ as valuation rings. Ostrowski's theorem gives the places of $\mathbb{Q}$ as the Archimedean place and the primes, the completions are $\mathbb{R}$ and the fields $\mathbb{Q}_p$, and the product formula holds over all places at once.

Every finite extension of $\mathbb{Q}$ is separable and simple, so it is $\mathbb{Q}(\alpha)$ for an algebraic $\alpha$; the algebraic closure $\overline{\mathbb{Q}}$ is countable and of infinite degree over $\mathbb{Q}$. The quadratic extensions are the fields $\mathbb{Q}(\sqrt d)$ for squarefree $d$, classified by the square classes $\mathbb{Q}^\times/(\mathbb{Q}^\times)^2$, and there are countably many of them; the cyclotomic fields $\mathbb{Q}(\zeta_n)$ are the abelian extensions generated by the roots of unity, with Galois group $(\mathbb{Z}/n\mathbb{Z})^\times$ and degree $\varphi(n)$, and by the Kronecker–Weber theorem they generate the maximal abelian extension of $\mathbb{Q}$. The field $\mathbb{Q}$ is rigid in the strongest sense: its only unital endomorphism is the identity, the only homomorphism of $\mathbb{Q}$ into itself that is not unital is the zero map, and its only derivation is zero.

Property	Value for $\mathbb{Q}$
Characteristic	$0$
Prime field	itself
Cardinality	$\aleph_0$
Orderings	exactly one, Archimedean, non-complete
Real closed	no
Algebraically closed	no; closure $\overline{\mathbb{Q}}$, countable
Unit group	$\mathbb{Z}/2 \oplus \bigoplus_p \mathbb{Z}$
Subrings	localizations $\mathbb{Z}_S$
Completions	$\mathbb{R}$, $\mathbb{Q}_p$ $(p$ prime$)$
Automorphisms	$\operatorname{Aut}(\mathbb{Q}) = 1$

Summary of Notation

Symbol	Meaning
$\mathbb{Z}$	Integers
$\mathbb{Q} = \operatorname{Frac}(\mathbb{Z})$	Rational numbers
$\mathbb{Q}^\times$	Nonzero rationals
$v_p$	$p$-adic valuation
$\lvert \cdot \rvert_p$, $\lvert \cdot \rvert_\infty$	$p$-adic and usual absolute values
$\operatorname{sgn}(a)$	Sign of a nonzero rational
$\mathbb{Z}_S$, $\mathbb{Z}_{(p)}$	Localizations of $\mathbb{Z}$
$\overline{\mathbb{Q}}$	Algebraic numbers, the algebraic closure; $[\overline{\mathbb{Q}}:\mathbb{Q}]$ is infinite
$\mathbb{R}$, $\mathbb{Q}_p$	The completions of $\mathbb{Q}$
$\operatorname{tr.deg}$	Transcendence degree
$\operatorname{Aut}(\mathbb{Q})$	Automorphism group, trivial
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$	Absolute Galois group of $\mathbb{Q}$
$\operatorname{End}(\mathbb{Q})$, $\operatorname{Der}(\mathbb{Q})$	Unital endomorphisms $\{\mathrm{id}\}$, all homomorphisms $\{0,\mathrm{id}\}$, derivations $0$
$(n,m) \sim (k,\ell)$	Equivalence relation defining $\mathbb{Q}$
$(\mathbb{Q}^\times)^2$	Squares, defining the square classes $\mathbb{Q}^\times/(\mathbb{Q}^\times)^2$
$\mathbb{Q}(\sqrt d)$	Quadratic field, $d$ squarefree
$\zeta_n$, $\varphi(n)$	Primitive $n$-th root of unity, Euler function
$\mathbb{Q}(\zeta_n)$	Cyclotomic field, Galois group $(\mathbb{Z}/n\mathbb{Z})^\times$