Algebraic number theory

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.

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 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.

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.

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 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.

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 of the rational numbers.

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 degree-two extension of $\mathbb{Q}$, necessarily of the form $\mathbb{Q}(\sqrt{d})$ for a square-free integer $d\ne 0,1$.

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.

The algebraic conjugates of an algebraic number are the roots of its minimal polynomial, equivalently its images under embeddings of its number field into an algebraic closure.

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.

In a Galois extension, the prime ideals above a fixed unramified prime have a common ramification index and residue degree, and the Galois group acts transitively on them.

A prime ideal of a number field splits completely in a finite extension of degree $n$ when it factors into $n$ distinct prime ideals, each of residue-field degree one. In a Galois extension, an unramified prime ideal splits completely exactly when its Frobenius conjugacy class is the identity class. The Chebotarev density theorem then gives these primes Dirichlet density $1/n$.

For a quadratic field of discriminant $D$ and a rational prime $p$, the ideal $(p)$ splits into two distinct prime ideals, remains prime, or ramifies according as the quadratic character ${\chi }_D(p)$ is $1$, $-1$, or $0$. The corresponding prime-ideal norms are $(p,p)$, $p^2$, and $p$, respectively.

A prime $p$ ramifies in a number field when its prime-ideal factorization contains a factor with exponent greater than one.

A finite extension of number fields is unramified everywhere when every finite prime ideal is unramified. If infinite places are included in the convention, one also requires every real place to remain real.

If $p\equiv 11(\mathrm{mod}12)$ and $K=\mathbb{Q}(\sqrt{-p})$, then ${\mathcal{O}}_K=\mathbb{Z}[(1+\sqrt{-p})/2]$. The generator's minimal polynomial reduces modulo $3$ to $T(T-1)$, so $(3)$ splits into two distinct prime ideals of norm three.

The discriminant of a biquadratic field is the product of the fundamental discriminants of its three quadratic subfields.

An element $x$ is integral over $\mathbb{Z}$ exactly when it acts stably on some faithful nonzero finitely generated $\mathbb{Z}$-module containing one.

A ring generated over $\mathbb{Z}$ by finitely many algebraic integers is a finitely generated $\mathbb{Z}$-module.

If $B$ is integral over $A$ and an element is integral over $B$, then it is integral over $A$.

An integral basis is a $\mathbb{Z}$-basis of the ring of integers of a number field.

If $m>1$ is a square-free integer, $3\nmid m$, and $m\not\equiv \pm 1(\mathrm{mod}9)$, then $1,\sqrt[3]{m},\sqrt[3]{m^2}$ is an integral basis of the pure cubic number field, with field discriminant $-27m^2$. The discriminant-index formula for an integral lattice restricts a possible index to primes dividing $3m$. At primes dividing $m$ the different exponent is $2$ by tame ramification. A shifted Eisenstein polynomial gives total wild ramification at $3$, so its different exponent is at least $3$. These exponents exhaust the polynomial discriminant and force index one.

The discriminant of an integral full-rank lattice is the field discriminant multiplied by the square of its index in the ring of integers. A squarefree lattice discriminant therefore forces index one.

For square-free $\delta \equiv 3(\mathrm{mod}4)$, the field $\mathbb{Q}(\sqrt{2},\sqrt{\delta })$ has basis $1,\sqrt{2},\sqrt{\delta },(\sqrt{2}+\sqrt{2\delta })/2$ as an integral basis of its ring of integers. Relative traces force all rational coefficients to be half-integers; the norm to $\mathbb{Q}(\sqrt{\delta })$ forces the constant and $\sqrt{\delta }$ coefficients to be integers and the two other numerators to have equal parity. The half-sum is integral because it satisfies $X^4-(1+\delta )X^2+(\delta -1{)}^2/4=0$.

For elements ${\alpha }_1,\dots ,{\alpha }_n$ of a degree-$n$ number field with embeddings ${\sigma }_1,\dots ,{\sigma }_n$, their discriminant isIt is nonzero exactly when the elements form a $\mathbb{Q}$-basis.

The field discriminant is the discriminant of any integral basis and controls the index of every suborder.

A ramified prime makes the finite algebra ${\mathcal{O}}_K/p{\mathcal{O}}_K$ have a nonzero nilpotent element. Multiplication by its product with any other element is nilpotent and has trace zero, so the reduced trace pairing is degenerate. Therefore the field discriminant is divisible by $p$. This proves finiteness of the ramified rational primes without requiring explicit prime factorization in the field.

Let ${\omega }_1,\dots ,{\omega }_n$ be an integral basis and let ${\alpha }_j={\sum }_i{a}_{ij}{\omega }_i$ be a rational basis of algebraic integers, with integer matrix $A=(a_{ij})$. Their trace Gram matrices satisfysoThe absolute discriminant is minimized exactly when $A$ is unimodular, equivalently when the ${\alpha }_i$ form an integral basis.

The polynomial $X^3-5X+8$ is irreducible over $\mathbb{Q}$. Forone hasThe field has ring of integers $\mathbb{Z}[\beta ]$ and field discriminant $-307$.

The elementsform an integral basis. Relative to this basis, multiplication by $\beta$ has integer matrixThe basis discriminant is one quarter of the discriminant $-4\cdot 307$ of $(1,\alpha ,{\alpha }^2)$, hence is the square-free integer $-307$.

For $g(X)=X^3-5X^2+11X-12$, the reductions areandBoth quadratic factors are irreducible, soand

The field norm $N_{L/K}(x)$ is the determinant of multiplication by $x$, equivalently the product of its conjugates when the extension is separable.

For a finite number field $L/\mathbb{Q}$ with a rational basis ${\omega }_1,\dots ,{\omega }_n$, the norm form is the homogeneous polynomial

If $x=y^2$ in a number field, then $N(x)=N(y{)}^2$; a norm that is not a square in the base field therefore obstructs square roots.

An algebraic integer whose absolute norm is $1$ or $-1$ is a unit, because its conjugate product supplies an integral inverse.

For a number field of signature $(r_1,r_2)$, the Minkowski embedding sends an element to its values under the $r_1$ real embeddings and one representative from each of the $r_2$ complex-conjugate pairs. The image of the ring of integers is a full lattice in ${\mathbb{R}}^{r_1}\times {\mathbb{C}}^{r_2}$.

Let $j(\mathfrak{b})$ be the Minkowski embedding of a number field applied to a nonzero fractional ideal. Under the metric $|x{|}^2={\sum }_{v\text{ real}}{x}_v^2+2{\sum }_{v\text{ complex}}|x_v{|}^2$, its covolume is $\sqrt{|D_K|}N(\mathfrak{b})$. For ordinary coordinate Lebesgue measure $\prod d{x}_v\prod d\mathrm{Re}{z}_{v}d\mathrm{Im}{z}_v$, its covolume is instead $2^{-r_2}\sqrt{|D_K|}N(\mathfrak{b})$. Here $D_K$ is the field discriminant, and the absolute norm of a fractional ideal is positive and multiplicative. The formula follows by taking the embedding determinant of an integral basis, then using the index of an integral ideal and scaling to handle a fractional ideal.

A nonzero algebraic integer whose conjugates all have modulus at most one is a root of unity. Its norm forces every conjugate to have modulus one, and bounded integral coefficient vectors show that two positive powers coincide.

If $\Lambda \subset {\mathbb{R}}^n$ is a full lattice and a measurable convex centrally symmetric set $C$ has volume greater than $2^n\mathrm{covol}(\Lambda )$, then $C$ contains a nonzero lattice point.

For a real quadratic field $K$ with embeddings ${\sigma }_1,{\sigma }_2$, the imageis a lattice in ${\mathbb{R}}^2$ of covolume $\sqrt{|d_K|}$, and the product of the two coordinates is the field norm.

Let $K=\mathbb{Q}(\sqrt{-d})$ for square-free $d>0$ with $d\not\equiv 3(\mathrm{mod}4)$, so ${\mathcal{O}}_K=\mathbb{Z}[\sqrt{-d}]$. An integral ideal $I$ is a lattice in $\mathbb{C}\simeq {\mathbb{R}}^2$ of covolume $\sqrt{d}N(I)$. Applying the Minkowski convex body theorem to a disk gives a nonzero $\alpha \in I$ satisfying

If $f(X)=a_d{\prod }_{j=1}^d(X-{\alpha }_j)$, its Mahler measure is

Let an irreducible $f\in \mathbb{Z}[X]$ have $M(f)=2$ and a root outside the closed unit disk. Its leading coefficient must have absolute value one. The product $B$ of all roots outside the unit disk is then an algebraic integer with $|B|=2$; conjugate pairs make $B$ real, so $B=\pm 2$. The quotient of the norm of any root by $B$ is both rational and an algebraic integer, and its absolute value is at most one. It is therefore $\pm 1$, so every root has absolute norm two.

For every complex polynomial $P$,This follows from Jensen's formula, or directly by bounding $|P(z)|$ by its polynomial length on the unit circle and using the integral formula for the Mahler measure.

A place of a number field $K$ is an equivalence class of absolute values on a field. Its Archimedean places come from real embeddings and conjugate pairs of complex embeddings. Its non-Archimedean places correspond to nonzero prime ideals of the ring of integers of a number field.

Such a field embedding pulls the p-adic absolute value back to a finite place of a number field above $p$. It extends to an embedding of the corresponding completion, whose image is a finite extension of the P-adic numbers.

Finite places of a number field above $p$ correspond to absolute Galois group orbits of embeddings of a number field into a p-adic algebraic closure. The absolute value is invariant under the group. Conversely, embeddings inducing the same place extend to embeddings of the same local completion and are conjugate.

A real embedding $K\hookrightarrow \mathbb{R}$ gives a real Archimedean place of local degree $1$. A conjugate pair of complex embeddings $K\hookrightarrow \mathbb{C}$ gives a complex Archimedean place of local degree $2$.

A nonzero element of a number field is totally positive when its image is positive under every real field embedding. In a totally real number field, the principal fractional ideals having a totally positive generator form the subgroup used to define the narrow ideal class group. Signs of different generators are related by the unit signature map.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index of a prime ideal $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

For a place $v$ of a number field $K$, the local degree is $d_v=[K_v:{\mathbb{Q}}_v]$. Thus $d_v=1$ at a real place, $d_v=2$ at a complex place, and $d_v=e_v{f}_v$ at a finite place over $p$. For each rational place, the local degrees above it sum to $[K:\mathbb{Q}]$.

A height function measures the arithmetic complexity of an algebraic number or a point of an arithmetic variety.

For an algebraic number $\alpha$ in a number field $L$, the absolute multiplicative Weil height isIt is independent of the field $L$ containing $\alpha$, satisfies $H({\alpha }^n)=H(\alpha {)}^{|n|}$, and obeys $H(\alpha +\beta )\le 2H(\alpha )H(\beta )$.

For coprime integers $p,q$ with $q>0$,This follows directly from the real and p-adic absolute values, or from the height-Mahler measure formula for the primitive polynomial $qX-p$.

The absolute logarithmic Weil height is $h(\alpha )=\log H(\alpha )$.

For a nonzero vector $\mathbf{a}=(a_0,\dots ,a_n)$ over a number field $K$, its absolute projective height isThe product formula makes this unchanged by multiplying all coordinates by the same nonzero scalar. The projective height of a linear form is the projective height of its coefficient vector.

The naive height of a polynomial is the largest absolute value of its coefficients. For a primitive polynomial in $\mathbb{Z}[X_1,\dots ,X_n]$, this equals the projective height of its coefficient vector.

If the primitive minimal polynomial of an algebraic number $\alpha$ has degree $d$, then

Let $P,Q\in \mathbb{Z}[X_1,\dots ,X_k]$ have degree at most $n_j$ in $X_j$. Thenwhenever the quotient is defined. At each non-Archimedean place, the integral coefficients contribute at most one; at the Archimedean places, the triangle inequality contributes the polynomial length. Multiplying these local estimates and using the product formula gives the claim.

If $\gamma \ne 0$ is algebraic of degree at most $D$, then every embedding satisfiesIndeed, $H({\gamma }^{-1})=H(\gamma )$ and the contribution of one Archimedean place to the height bounds $|\gamma {|}^{-1/D}$.

There are only finitely many algebraic numbers of bounded degree and bounded Absolute multiplicative Weil height. The finite set can be enumerated effectively from the bounded coefficients of their primitive minimal polynomials.

For a nonzero element $x$ of a number field $L$, normalized absolute values satisfyFor an algebraic integer $x$, factor the principal ideal $(x)$ into prime ideals. Its ideal norm is both the product of the finite-place contributions with inverse exponent and the absolute value of the product of its Archimedean conjugates. Equating the two expressions proves the formula for algebraic integers, and writing an arbitrary $x\in L^{\times }$ as a quotient proves the general case.

The product formula uses finite-place factors given by ideal norms, the ordinary modulus at real embeddings, and the modulus squared at complex places. The complex square accounts for the pair of conjugate embeddings. Equivalently ordinary complex modulus can be used with weight two in the product. These normalizations cancel the absolute global field norm.

If a number field has $r$ real embeddings and $s$ conjugate pairs of complex embeddings, thenwhere $\mu (K)$ is the finite group of roots of unity in $K$.

In a real quadratic number field, a fundamental unit $ϵ>1$ generates the infinite cyclic factor of its unit group, so every unit is $\pm {ϵ}^n$. Existence follows from the Dirichlet unit theorem; choosing the positive generator of the rank-one logarithmic lattice gives this normalization.

If a squarefree positive $d$ has a prime divisor $p\equiv 3(\mathrm{mod}4)$, a unit of norm $-1$ in $\mathbb{Q}(\sqrt{d})$ would give $a^2-d{b}^2=-4$ for integers $a,b$. Modulo $p$ this makes $-1$ a quadratic residue, which is impossible. The obstruction is sufficient, not a full criterion for negative-norm units.

For $K=\mathbb{Q}(\sqrt{5})$,Writing $\varphi =(1+\sqrt{5})/2$, the full unit group is

The unit $\epsilon =1+\sqrt{2}$ has norm minus one and is the least unit greater than one in the positive real embedding. Dirichlet unit theorem therefore gives

Reduction modulo $(3+\sqrt{2})$ identifies $\mathbb{Z}[\sqrt{2}]/(3+\sqrt{2})$ with ${\mathbb{F}}_7$, sends $\sqrt{2}$ to $4$, and sends $\epsilon =1+\sqrt{2}$ to $5$. Since $5$ has order six and $5^3=-1$ in ${\mathbb{F}}_7^{\times }$, the kernel on units is the infinite cyclic group

If $K$ is real quadratic and $L/K$ is a number-field extension with ${\mathcal{O}}_L^{\times }/{\mathcal{O}}_K^{\times }$ finite, then the unit ranks agree. If $d=[L:K]$ and $L$ has signature $(r_1,r_2)$, this forcesFor a proper extension the only possibility is $d=2$ and signature $(0,2)$. For $K=\mathbb{Q}(\sqrt{5})$, the example $L=K(i)$ has this signature and a nontrivial finite quotient because $i$ is a new root of unity.