Algebraic number theory

Algebraic number theory studies integer factorization, divisibility, prime ideals, and units after extending the rational numbers to a number field. Its basic objects are a number field $K$ and its ring of integers ${\mathcal{O}}_K$: the counterparts of $\mathbb{Q}$ and $\mathbb{Z}$. The field norm turns multiplication in $K$ into multiplication of rational numbers, and takes algebraic integers to integers. These connections let linear algebra and Euclidean geometry answer arithmetic questions.^[ref]

A first example is $K=\mathbb{Q}(\sqrt{2})$ with ring of integers $\mathbb{Z}[\sqrt{2}]$. The integer $7$ acquires the factorizationEach factor has field norm $7$. Neither is a unit, and each is an irreducible element: any factorization into two nonunits would factor the prime number $7$ into two integers of absolute value greater than one. Arithmetic in a larger ring thus reveals structure invisible in ordinary integer factorization.

There is also a complication: unique factorization of elements can fail, as in the failure of unique factorization in the ring of integers generated by square root of minus five. The general replacement is prime ideal factorization. The ideal class group measures the obstruction to making every ideal a principal ideal.

The historical motivation is visible in Richard Dedekind's original 1877 exposition: he begins with ordinary integer factorization, passes to Gaussian integers, and then uses a quadratic example to explain why ideals are needed. His later section establishes unique factorization of ideals in a number field. This is a direct record of the development of the theory, rather than a claim that all its ideas originated in one year.^[ref][ref]

Elliptic units are algebraic units constructed by evaluating suitably normalized elliptic functions at CM torsion points and taking products or norms into ray class fields. Their distribution relations make them norm compatible in CM division towers. They generate an Euler system of elliptic units and their local Coleman power series for a CM formal group have logarithmic derivatives encoding critical Hecke L-function values.

Elliptic-unit distribution gives norm-compatible units indexed by auxiliary squarefree ray moduli, with norm factor $1-{\mathrm{Frob}}_{\mathfrak{q}}^{-1}$ at a newly added prime. Kolyvagin derivative operators turn these units into classes with controlled valuations, and suitable auxiliary primes chosen using Chebotarev density theorem bound class-group characteristic ideals. The construction supplies the difficult divisibility in the CM Iwasawa main conjecture.

A Hecke character of a number field is a continuous character of its idèle class group. Its finite conductor records the local unit subgroups on which it is trivial. As an ideal character it is defined on fractional ideals prime to that conductor, with a prescribed rule on principal ideals coming from its archimedean character. A CM elliptic curve yields an algebraic Hecke character with one CM embedding in its infinity type of a Hecke character.

Suppose $F$ contains the CM field $K$ and all CM endomorphisms are defined over $F$. CM reciprocity compares the torus with its idele-transformed lattice and gives $a(s)\in K^{\times }$ for finite ideles. Then ${\psi }_E(s)=a(s_f)/N_{F/K}(s{)}_{\infty }$. On principal ideles, $a(b_f)=N_{F/K}(b)$, so the character descends to the idèle class group. At a good prime its ideal value reduces to the Frobenius isogeny. If the original field omits $K$, construct the character over $FK$ and induce the corresponding representation down to the original field.

For $E:y^2=x^3-x$ over $\mathbb{Q}(i)$, the Hecke character has conductor $(1+i{)}^3$ and ideal infinity type $(1,0)$. Its value on an ideal prime to $2$ is its unique primary Gaussian integer generator. Rational $(1+i{)}^3$-torsion fixes the unit ambiguity of the Frobenius value. At a split prime $\mathfrak{p}$, its value reduces to the degree-$p$ Frobenius isogeny; at an inert rational prime $p$, its value is $-p$, reducing to the $p^2$-power Frobenius isogeny.

For an algebraic Hecke character, the character on archimedean ideles is a product of integral powers of embeddings and their complex conjugates. In the convention $\psi ((a))=a$ for primary $a$ over an imaginary quadratic field, the ideal character has type $(1,0)$ and the idele character has archimedean component $z^{-1}$. Distinguishing these conventions avoids inverse errors in CM Galois representations.

The factorization of a defining polynomial modulo a prime, provided the index is coprime to that prime, determines the prime ideals, ramification indices and residue degrees in the corresponding ring of integers.

For an algebraic integer $\alpha$ generating a number field, the polynomial discriminant of its minimal polynomial $f$ equals the discriminant of elements of a number field for its power basis. A squarefree reduction modulo $p$ has nonzero resultant with its derivative, so $p\nmid \mathrm{disc}(f)$. The discriminant-index formula for an integral lattice, $\mathrm{disc}(f)=[{\mathcal{O}}_K:\mathbb{Z}[\alpha ]{]}^2{d}_K$, proves the assertion.

// Target: arithmetic.bigb

For an integral generator $\alpha$ of a number field, let $A=\mathbb{Z}[\alpha ]$ and suppose $p$ is coprime to $[{\mathcal{O}}_K:A]$. Localizing at $p$ makes $A$ and the ring of integers equal, so reduction gives the displayed quotient ring. If $\overline{m}=\prod g_i^{e_i}$ with distinct monic irreducible polynomials, the Chinese remainder theorem gives local factors ${\mathbb{F}}_p[X]/(g_i^{e_i})$. Their residue fields have degrees $\mathrm{deg}{g}_i$ and their maximal ideals have nilpotence indices of an ideal $e_i$. Localizing the Dedekind domain ${\mathcal{O}}_K$ at the corresponding prime ideals gives discrete valuation rings, where this same nilpotence index of an ideal is the ramification index of a prime ideal. Thus $(p)=\prod (p,g_i(\alpha ){)}^{e_i}$ and each residue degree is $\mathrm{deg}{g}_i$.

The Leopoldt conjecture says that the diagonal image of the global units of a number field in its local $p$-adic unit groups has ${\mathbb{Z}}_p$-rank $r_1+r_2-1$. A defect would measure a loss of rank after $p$-adic completion. It is a theorem for abelian extensions of $\mathbb{Q}$.

The Leopoldt conjecture holds for every finite abelian extension of $\mathbb{Q}$. The proof uses linear independence of $p$-adic logarithms of algebraic numbers. In cyclotomic towers it supplies the ranks of the global-unit terms in the class-field unit sequence; it is not needed for the unramified Iwasawa torsion theorem in arbitrary towers.

Iwasawa theory studies arithmetic objects in infinite towers of number fields, often Zp-extensions. Passing to inverse limits turns growing ideal class groups, unit groups and Galois groups into modules over an Iwasawa algebra. Their algebraic ranks and characteristic ideals organize the growth and relate it to p-adic L-functions.

The classical Iwasawa main conjecture, now a theorem over $\mathbb{Q}$, identifies characteristic ideals of appropriate cyclotomic Iwasawa modules with ideals generated by p-adic L-function power series. For a nontrivial even character in the p-ramified convention, ${\mathrm{char}}_{\Lambda }{X}_{\chi }=(g_{\chi })$ with $g_{\chi }((1+p{)}^{1-s}-1)=L_p(\chi ,s)$. The Kummer reflection in Iwasawa theory accounts for the reflected odd class-group convention.

In the real cyclotomic tower, the class-field unit sequence gives $0\to E/C\to U/C\to X\to Y\to 0$, where $X$ is the p-ramified Iwasawa module and $Y$ the unramified one. The modules are torsion over each one-variable character component. Splitting the four-term sequence into two short exact sequences proves the displayed identity by multiplicativity of characteristic ideals. The Iwasawa theorem on local cyclotomic units computes $\mathrm{char}(U/C)$ analytically. Thus the main conjecture for $X$ is equivalent to $\mathrm{char}(E/C)=\mathrm{char}(Y)$; local interpolation alone does not prove this remaining global equality.

For the real p-power cyclotomic tower, take norm limits of local principal units and closed cyclotomic-unit subgroups. The corrected Coleman logarithm identifies $U_{\infty }^1$ with the Iwasawa algebra of $G={\mathbb{Z}}_p^{\times }/\{\pm 1\}$. On a symmetric cyclotomic unit it gives $([b]-[a]){\zeta }_p$. A topological generator generates the cyclotomic-unit limit and the augmentation ideal, yielding the quotient description. This local calculation does not alone identify the characteristic ideal of the global Galois module.

At a good split prime, the one-prime CM division tower has a pro-$p$ Galois group isomorphic to ${\mathbb{Z}}_p$. Its prime-ramified Iwasawa module is torsion on each finite-character component. The main conjecture identifies its characteristic ideal with the corresponding elliptic-unit p-adic L-function. The class-field unit sequence and equality of characteristic ideals of global-units modulo elliptic-units and of class groups prove the identification. The cyclotomic Selmer formulation requires the appropriate CM twist and two-variable specialization; the one-prime tower is not the cyclotomic tower.

The Mazur-Wiles theorem proves the classical Iwasawa main conjecture for abelian extensions of $\mathbb{Q}$. Its construction of class fields uses Galois representations associated with modular forms, congruences with Eisenstein series and the Eisenstein ideal.

An Iwasawa module is a module over an Iwasawa algebra. In arithmetic towers it usually comes from a norm inverse limit of units or class groups, or from a compact Galois module for a maximal abelian pro-$p$ extension. Specify which extension is allowed to ramify: the p-ramified Iwasawa module and the unramified Iwasawa module have different ranks.

Let $M_{\infty }$ be the maximal abelian pro-$p$ extension of $F_{\infty }$ unramified away from primes over $p$. Its Galois group is the p-ramified Iwasawa module. In the cyclotomic tower of $\mathbb{Q}({\mu }_p)$ for odd $p$, its Iwasawa-module rank is $(p-1)/2$. It is therefore quite different from the torsion unramified Iwasawa module.

Passing to norm limits in Artin reciprocity gives an exact sequence connecting closures of global units, local pro-$p$ units at primes over $p$, the p-ramified Iwasawa module and the unramified Iwasawa module. It explains arithmetic Iwasawa-module ranks by comparing local and global unit ranks. Roots of unity and the precise splitting conventions must be retained when seeking exact integral statements.

A Coleman power series encodes a norm-compatible sequence of local units in a cyclotomic tower by a single integral power series evaluated at ${\zeta }_{p^n}-1$. A logarithmic derivative and trace correction give a measure, relating the local images of cyclotomic units to p-adic L-functions.

For odd $p$ and a norm-compatible sequence of local units in ${\mathbb{Q}}_p({\mu }_{p^{n+1}})$, there is a unique $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ interpolating all its entries, and the Coleman norm operator fixes it. To construct it, choose integral unit polynomials $F_j$ interpolating entry $u_j$, and take a convergent subsequence of $N^j{F}_{2j}$ in the compact $(p,T)$-adic unit space. The congruence $NF\equiv F(\mathrm{mod}p)$ and Coleman norm contraction show that its evaluation at any fixed level tends to the desired entry. Uniqueness follows from the Weierstrass preparation theorem: a nonzero integral power series has only finitely many zeros in the open unit disk. The norm relation on evaluations then proves norm fixation. Principal units correspond to $f_u(0)\equiv 1(\mathrm{mod}p)$.

Let $V_{\infty }$ be the norm limit of local principal units in the $p$-power cyclotomic field tower, for odd $p$. The Coleman power series $f_u$ gives the Frobenius-corrected logarithm $p^{-1}\log (f_u^p/\varphi f_u)$, whose inverse Amice transform is supported on ${\mathbb{Z}}_p^{\times }$. This is the middle map in the displayed exact sequence; the last map is the first moment $\mu \mapsto \int xd\mu$. Its kernel consists of norm-compatible $p$-power roots of unity. The logarithmic difference equation has one resonant degree, namely degree one in $\log (1+T)$, accounting for the last term. Complex conjugation acts by $-1$ on both copies of ${\mathbb{Z}}_p(1)$. Taking its even part therefore identifies the local principal-unit limit in the real tower with ${\mathbb{Z}}_p[[{\mathbb{Z}}_p^{\times }/\{\pm 1\}]]$.

Such a sequence $u=(u_n)$ satisfies $N_{n/n-1}{u}_n=u_{n-1}$ at every successive layer. The inverse limit uses the field norm, rather than inclusion of units from the lower field. In a p-power cyclotomic tower a Coleman power series interpolates the sequence, and its Coleman norm operator fixes the interpolating series.

For a norm-compatible sequence of local units, its Coleman power series $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ defines these integral additive homomorphisms. In the formal coordinate $z=\log (1+T)$ they are the derivatives of $\log f_u(e^z-1)$; the constant term of the logarithm is irrelevant. A Galois element with cyclotomic value $a$ multiplies the kth derivative by $a^k$. This is the multiplicative-formal-group version of the Coates–Wiles homomorphism.

The determinant norm for the finite free extension $R/\phi (R)$ defines $\mathcal{N}$ on units. After adjoining the p-th roots of unity, the norm is the displayed product of conjugates. A Coleman power series of a norm-compatible unit sequence satisfies $\mathcal{N}f=f$. For symmetric cyclotomic units, the product relation follows directly from ${\prod }_{{\xi }^p=1}(1-{\xi }^a{X}^a)=1-X^{ap}$ when $p\nmid a$.

Every unit $\overline{f}\in {\mathbb{F}}_p[[T]{]}^{\times }$ has a unique Coleman norm operator-fixed lift to ${\mathbb{Z}}_p[[T]{]}^{\times }$. Choose any lift $f$ and use $Nf\equiv f(\mathrm{mod}p)$. The consecutive quotients of the iterates are $N^n(Nf/f)\in 1+p^{n+1}R$ by Coleman norm contraction, so they converge coefficientwise uniformly in the p-adic integers. Their limit is norm-fixed by continuity. Two such lifts have norm-fixed quotient in $1+pR$ and therefore coincide. Uniqueness also makes the lifting map multiplicative and independent of the initial lift.

Let $W$ be the subgroup fixed by the Coleman norm operator. The Coleman norm-fixed lift gives the unique factorization $f=w(\overline{f})v$, with $v\in 1+pR$. Multiplication consequently gives the displayed direct product of groups, while reduction identifies $W$ with ${\mathbb{F}}_p[[T]{]}^{\times }$. This is a decomposition of unit groups, not an identification of a power-series ring with a product of groups.

For odd $p$, the Coleman norm operator satisfies the displayed inclusion for every $k\ge 1$. Write $M_h$ for multiplication by $h$ on the finite free module $R$ over $\varphi (R)$. Its trace is $p\varphi (S(h))$, where $S$ is the Coleman trace operator. In $\det (I+p^k{M}_h)$ the linear term is therefore divisible by $p^{k+1}$, and all higher terms are divisible by $p^{2k}$. Congruence reflection for Frobenius substitution gives the assertion. A norm-fixed element of $1+pR$ lies in $1+p^{k}R$ for every $k$ and hence equals $1$.

On $R={\mathbb{Z}}_p[[T]]$, this injective substitution reduces modulo $p$ to $f(T)\mapsto f(T^p)$. The ring is a free module of rank $p$ over $\phi (R)$, with basis $1,(1+T),\dots ,(1+T{)}^{p-1}$. Prove this first modulo $p$, where the change of basis from $1,T,\dots ,T^{p-1}$ is triangular and invertible, and then lift coefficients successively modulo powers of $p$.

For $R={\mathbb{Z}}_p[[T]]$ and $\varphi (f)=f((1+T{)}^p-1)$, reduction modulo $p$ is the injective substitution $\overline{f}(T)\mapsto \overline{f}(T^p)$. Thus $\varphi (f)\in pR$ implies $f\in pR$. Dividing by $p$ and repeating proves the displayed equality for every $k\ge 1$. In particular, a ring congruence of Frobenius substitutions on cyclotomic power series reflects back to the original formal power series.

In the unique decomposition $f={\sum }_{i=0}^{p-1}(1+T{)}^i\phi (f_i)$, define $\psi (f)=f_0$. Averaging translates over the p-th roots of unity annihilates every nonzero basis index and retains the zeroth, proving the displayed identity and its integrality. Thus $\psi \phi =1$ and $\psi (\phi (g)f)=g\psi (f)$. Under the Amice transform, $\psi =0$ characterizes measures supported on ${\mathbb{Z}}_p^{\times }$.

For a height-one CM elliptic curve formal group and compatible ${\mathfrak{p}}^n$-division parameters $t_n$, an integral Coleman series interpolates a norm-compatible sequence of local units $u_n$. Its norm relation is multiplication of its translates by the $\mathfrak{p}$-torsion, with coefficient Frobenius where required by the unramified base. Coates–Wiles homomorphisms and the Frobenius-corrected logarithm turn this local series into critical values and measures.

Assume $K_{\mathfrak{p}}={\mathbb{Q}}_p$, $p>2$, $f(t)\in {\mathbb{Z}}_p[[t]{]}^{\times }$, and $[\pi ]t\equiv t^p(\mathrm{mod}p)$. Then the displayed ratio lies in $1+p{\mathbb{Z}}_p[[t]]$, so its logarithm divided by $p$ is integral. A formal-group isomorphism over the completed unramified coefficient ring transfers it to an integral Amice transform. A norm relation makes the corresponding p-adic measure supported on units. Its $k$th moment has the Euler correction $1-{\pi }^k/p$.

Let $R={\mathbb{Z}}_p[[T]]$, $p$ odd, and $\varphi (T)=(1+T{)}^p-1$. An element $g\in R$ is the Frobenius-corrected logarithm of a unit $f$ with $f(0)\in 1+p{\mathbb{Z}}_p$ if and only if $g^{\prime }(0)=0$. Necessity follows by differentiating at zero, using ${\varphi }^{\prime }(0)=p$. For sufficiency put $H=\exp (pg)\in 1+pR$ and solve $f^p=H\varphi (f)$. Choose $f_0=\exp (p{g}_0/(p-1))$. The linear equation reduces to $H_1=0$, so $f_1$ is free. In degree $n\ge 2$ the unknown $f_n$ has coefficient $p{f}_0^{p-1}(1-p^{n-1})$. The remaining terms are divisible by $p$ because reduction of any integral truncated $f$ satisfies $f^p\equiv \varphi (f)(\mathrm{mod}p)$. Division gives a unique integral $f_n$. Two lifts differ by $(1+T{)}^c$, $c\in {\mathbb{Z}}_p$, as follows from the logarithmic equation in the coordinate $z=\log (1+T)$. If also $Sg=0$, the Coleman trace operator gives $\log (Nf/f)=pSg=0$. As $Nf/f\in 1+pR$, injectivity of the p-adic logarithm implies $Nf=f$.

The $k$th logarithmic derivative in the additive formal logarithm coordinate defines the Coates–Wiles homomorphism. Products of local units become sums under this map. For elliptic units, differentiating the underlying complex elliptic function expresses the result in terms of critical Hecke L-function values divided by a CM period.

For a Zp-extension, let $L_{\infty }$ be the maximal unramified abelian pro-$p$ extension of $F_{\infty }$. Its Galois group is a compact Iwasawa module, identifiable by Artin reciprocity with the inverse limit of the $p$-primary ideal class groups of finite layers under norms.

The unramified Iwasawa module is finitely generated torsion over the Iwasawa algebra of a Zp-extension, for every Zp-extension of a number field. After a finite shift all ramified primes are totally ramified and their number $s$ is constant. Class field theory bounds finite-layer coinvariant modules by ${\mathrm{rank}}_{{\mathbb{Z}}_p}{Y}_{{\Gamma }_n}\le s-1$. The Compact Nakayama lemma proves finite generation, and a positive Iwasawa-module rank would force ranks at least $r{p}^n$, a contradiction. No Leopoldt conjecture is required.

A finitely generated module over $\Lambda ={\mathbb{Z}}_p[[T]]$ is pseudo-isomorphic towhere the $f_j$ are irreducible distinguished polynomials. Height-one localizations are discrete valuation rings, which give the elementary divisors; the remaining errors are supported only at the maximal ideal and are finite. This is a finite-error classification, not necessarily an actual direct-sum decomposition.

Assume the invariant and coinvariant groups are finite and $f_M$ generates the characteristic ideal over ${\mathbb{Z}}_p[[S]]$, $S=\gamma -1$. The structure theorem and multiplicativity of finite Euler characteristics prove the formula; finite module errors have Euler characteristic one. If $f_M\mid f_N$ and the two Euler characteristics agree, the quotient has p-adic-unit constant term and is therefore a unit power series. Thus divisibility plus this finite arithmetic comparison upgrades to equality of characteristic ideals.

For a finitely generated torsion Iwasawa module, its characteristic ideal is the product of the height-one prime ideals raised to the lengths of the corresponding localizations. In the elementary-divisor description it is generated by $p^{\sum a_i}\prod f_j^{b_j}$. It is unchanged by finite modules and multiplicative in short exact sequences of torsion modules.

For a finitely generated torsion Iwasawa module, this is a generator of its characteristic ideal, unique up to a unit of the formal power series ring. It may be chosen as a power of $p$ times a product of distinguished polynomials. The series itself and the ideal it generates are different objects; changing a generator by a unit preserves the ideal. When the finite Euler characteristic of a one-variable Iwasawa module is defined, the inverse p-adic absolute value of its constant term gives that invariant.

The invariants $\mu ,\lambda$ of a finitely generated torsion Iwasawa module measure its $p$-power elementary factors and distinguished-polynomial factors. Vanishing of $\mu$ means that the module is finitely generated over ${\mathbb{Z}}_p$, up to finite error. It does not mean that the module is finite: a factor $\Lambda /(T)$ has $\mu =0$ and is ${\mathbb{Z}}_p$.

For the finite layers of a Zp-extension of a number field, the p-primary ideal class groups eventually satisfy the displayed formula with integer invariants $\mu ,\lambda \ge 0$ and $\nu \in \mathbb{Z}$. The first two are the Iwasawa invariants of the unramified Iwasawa module. The Iwasawa module structure theorem and control of the finite-layer norm and ramification terms give the result. For a cyclotomic extension of an abelian field, the Ferrero-Washington theorem gives $\mu =0$. This growth theorem records two invariants; it is weaker than identifying the full characteristic ideal in the Iwasawa main conjecture.

For a finite abelian extension of $\mathbb{Q}$, the unramified Iwasawa module of its cyclotomic Zp-extension has $\mu =0$. This removes $p$-power elementary factors from its characteristic ideal. The theorem does not assert $\mu =0$ for arbitrary noncyclotomic Zp-extensions.

A Zp-extension is an infinite Galois extension of a number field whose Galois group is topologically isomorphic to the additive p-adic integers. There is a unique intermediate field of degree $p^n$ for every $n\ge 0$, corresponding to $p^n{\mathbb{Z}}_p$. The finite layers form a tower with cyclic successive degree $p$ extensions.

These are units that lie in every finite-layer norm image, using the p-adic closures of the unit groups. Compactness makes this intersection the image of the norm inverse limit. Local and global universal norm subgroups need not coincide. In the cyclotomic main-conjecture proof, the finite defect of global units from universal norms is related by class field theory to invariants of the unramified module.

At least one finite prime ramifies in a Zp-extension; otherwise the entire tower would lie in a finite Hilbert class field. Ramification can occur only over $p$: local class field theory makes inertia away from $p$ an image of a unit group with finite maximal pro-$p$ quotient, whereas ${\mathbb{Z}}_p$ has no nontrivial finite subgroup. Nonzero inertia is open in ${\mathbb{Z}}_p$, so it becomes total after passing to a sufficiently high finite layer.

For odd $p$, remove the finite torsion subgroup ${\mu }_{p-1}$ from the Galois group of $\mathbb{Q}({\mu }_{p^{\infty }})/\mathbb{Q}$ by taking its fixed field. For $p=2$, take the fixed field of $\{\pm 1\}$ instead. The remaining Galois group is ${\mathbb{Z}}_p$, using the p-adic logarithm on $1+p{\mathbb{Z}}_p$ or $1+4{\mathbb{Z}}_2$. For a number field $K$, the cyclotomic Zp-extension is the compositum with this rational tower.

Given finitely many distinct places of a number field $v_1,\dots ,v_r$, the diagonal image of $K$ is dense in ${\prod }_i{K}_{v_i}$. One can therefore simultaneously prescribe finite congruences and real signs. This is useful in the ray class exact sequence and in finding representatives of an ideal class prime to a prescribed finite set of prime ideals.

At finitely many finite places of a number field, choose square-class parameters for the local unramified quadratic extensions. Local square classes are open, including at dyadic places by the strong form of Hensel lemma. Weak approximation for number fields gives a global parameter in all the prescribed local classes. Its quadratic extension is unramified of residue degree two at those places, so their extended prime ideals remain prime. If no finite place is prescribed, an auxiliary local nonsquare condition ensures the global extension is nontrivial.

The algebraic norm of an element of a finite field extension is the determinant of multiplication by that element. In a quadratic subring of $\mathbb{C}$, it is often $N(\alpha )=\alpha \overline{\alpha }$.

For algebraic elements $\alpha$ and $\beta$,This follows because multiplication by $\alpha \beta$ is the composition of multiplication by $\beta$ and multiplication by $\alpha$, and determinants are multiplicative.

For fixed degree and bounded absolute discriminant, there are only finitely many number fields up to isomorphism. In particular, only finitely many extensions of bounded degree can be unramified outside a fixed finite set of primes when their ramification exponents are bounded.

Fix a number field $K$, a positive integer $d$, and a finite set $S$ of its finite places. There are finitely many field extensions $L/K$ of degree at most $d$ unramified outside $S$, inside a fixed algebraic closure. To see this, view them as extensions of the rational numbers of bounded degree. Their ramified rational primes belong to a fixed finite set. At each of these primes, the different exponent is bounded in terms of the local degree and ramification index, giving a bound for the absolute field discriminant. The Hermite–Minkowski theorem gives finitely many fields up to isomorphism, and each has only finitely many embeddings into the fixed closure. This useful form of arithmetic compactness is stated and proved for $K=\mathbb{Q}$ in Andrew Sutherland's number-theory notes, Theorem 14.25; the preceding reduction gives the general case.

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.