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The Ostrowski theorem says that every nontrivial absolute value on a field defined on $\mathbb{Q}$ is equivalent either to the usual absolute value or to $|\cdot {|}_p$ for a unique prime $p$.

Let an absolute value on the number field $K$ extend $|\cdot {|}_p$. Its valuation ring determinesa prime ideal satisfying $\mathfrak{p}\cap \mathbb{Z}=(p)$. Conversely, each prime $\mathfrak{p}$ above $p$ defines the normalized absolute valuewhere $e_{\mathfrak{p}}=v_{\mathfrak{p}}(p)$. It restricts to $|\cdot {|}_p$. The correspondence between extensions and primes follows either from the valuation ring or from local factorization and extended absolute values; distinct primes give inequivalent valuations. Thus these $|\cdot {|}_{\mathfrak{p}}$ are exactly the extensions, up to equivalence.

For the tensor-product assertion, choose a primitive element of a field extension $\alpha$ for $K/\mathbb{Q}$, with minimal polynomial $f$. Because number fields are separable, over ${\mathbb{Q}}_p$ it factors into distinct irreduciblesindexed by the primes $\mathfrak{p}\mid p$. The Chinese remainder theorem givesThe $i$th factor is the completion of a number field at a prime ideal $K_{{\mathfrak{p}}_i}$. Under these identifications the isomorphism is the natural diagonal map $x\otimes a\mapsto (ax{)}_{\mathfrak{p}}$, proving the p-adic tensor decomposition of a number field

For the general result, choose logarithms of nonzero algebraic numbers ${\alpha }_1,\dots ,{\alpha }_s$ and algebraic coefficients ${\beta }_0,\dots ,{\beta }_s$, and putLetwhere $f_{{\alpha }_j}$ is the minimal polynomial and its $H$ is the naive polynomial height, and letThe general lower bound for a linear form in logarithms states that, if $\Lambda \ne 0$, thenwhere the effective constant $C$ depends only on $s$ and the degree of the number field generated by all the data.

For the improved homogeneous result, take $b_1,\dots ,b_s\in \mathbb{Z}$ andWith the same $A_j$, after ordering the terms setThe Baker lower bound for a homogeneous linear form in logarithms givesBoth constants are effective. The division by $\log A_s$ in $B^{\ast }$ is the improvement that matters when ${\alpha }_s$ has variable height.

Let $K$ be a number field. Its places consist of its real embeddings, conjugate pairs of complex embeddings, and the finite places associated with nonzero prime ideals $\mathfrak{p}$ of its ring of integers of a number field. At a real or complex place use the usual absolute value. If $\mathfrak{p}$ lies over the prime number $p$ with ramification index $e_{\mathfrak{p}}$, normalize its absolute value byThese normalizations extend the standard absolute values on $\mathbb{Q}$.

Write $d_v=[K_v:{\mathbb{Q}}_v]$ for the local degree of a place. Thus $d_v$ is $1$ at a real place, $2$ at a complex place, and $e_v{f}_v$ at a finite place. The Absolute multiplicative Weil height isThe product formula shows that this is unchanged when $K$ is replaced by a larger number field containing $\alpha$.

For a linear form $L=a_1{X}_1+\cdots +a_N{X}_N$ over a number field $K$, define its height to be the projective height of its coefficient vector:The product formula makes this independent of multiplying $L$ by a nonzero scalar.

The Siegel lemma says the following. Let $D=[K:\mathbb{Q}]$, let $L_1,\dots ,L_M$ be linear forms in $N$ variables with $DM<N$, and suppose $H_p(L_i)\le \mathcal{H}$ for every $i$, where $\mathcal{H}\ge 1$. Then there is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ annihilated by all the forms and satisfying

Choose a number field $L$ containing ${\alpha }_1,\dots ,{\alpha }_k$. At every place $v$ of $L$, putThe triangle inequality giveswhere $c_v=1$ at every non-Archimedean place because the coefficients are integers, while at an Archimedean place one may takeRaise these inequalities to the local weights and multiply over all places. The definition of the Absolute multiplicative Weil height and the product formula then giveThis is the height bound for a polynomial evaluation.

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.

Equivalence is taken among absolute values on a field, so changing a representative within one place does not change the associated topology.

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.

For a finite extension of number fields $L/K$ and a nonzero prime ideal $\mathfrak{p}$ of $K$, the primes of $L$ above $\mathfrak{p}$ correspond to the orbits ofAn embedding determines a prime by pulling back the local absolute value on a field. Embeddings with the same prime extend to embeddings of the same completion and are conjugate. If $L=K(\alpha )$, these orbits are the irreducible factors of its minimal polynomial over $K_{\mathfrak{p}}$.

Let $I$ be an integral ideal of an imaginary quadratic number field and $(e_1,e_2)$ an integral basis of $I$. Its normalized norm is a positive definite binary quadratic form. An unramified rational prime $p$ is represented by this form exactly when there is an integral prime ideal $P$ of norm $p$ with $[P]=[I{]}^{-1}$. Indeed, a representation gives an element $\alpha \in I$ with $N\alpha =pN(I)$, so $P=(\alpha )I^{-1}$ is integral of norm $p$. Conversely, $IP=(\alpha )$ with $\alpha \in I$ produces a representation by expressing $\alpha$ in the basis. Artin reciprocity then detects this ideal class by Frobenius in the Hilbert class field.

A principal fractional ideal of an integral domain $A$ with fraction field $K$ is $aA$ for a nonzero $a\in K$. Its generators differ by a unit in a ring in $A$. For a number field, taking $A={\mathcal{O}}_K$ and quotienting nonzero fractional ideals by principal fractional ideals gives the ideal class group; requiring congruences and signs of the generators instead gives a ray class group.

An idele obtained by mapping the same nonzero element of a number field into every completion. Only finitely many local valuations are nonzero, so this diagonal tuple belongs to the idele group.

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.

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.

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]

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 $\mathrm{deg}{g}_i$.

Suppose ${\alpha }^m=a\in {\mathbb{Q}}^{\times }$ in a number field and $\mathfrak{p}$ lies above $p$. Its normalized integral valuation satisfies $m{v}_{\mathfrak{p}}(\alpha )=e_{\mathfrak{p}/p}{v}_p(a)$. If $m$ and $v_p(a)$ are coprime, then $m\mid e_{\mathfrak{p}/p}$. In a degree-$m$ field the fundamental identity for prime decomposition forces a unique prime, ramification index $m$, and residue degree one. This detects total ramification of a prime even when the displayed defining polynomial is not Eisenstein at $p$.

// Target: algebra.bigb

Ramification is branching in a geometric or algebraic map, detected by multiplicity greater than one. For number fields, ramification of a prime means that some prime ideal occurs with exponent greater than one in the extended prime ideal. For a map of smooth algebraic curves, a local parameter pulls back to a parameter with order greater than one at a ramification point.

For an elliptic curve over a number field $K$, the Mordell-Weil theorem writes $E(K)$ as a finite torsion subgroup plus a free abelian group of finite rank of an abelian group $r$. This integer is the rank of the elliptic curve over $K$. It can increase under field extension. A two-isogeny descent supplies rank bounds from rational square-class images; the canonical height pairing can certify independence of rational points.

For an algebraic variety $X$ defined over a field $K$, a K-rational point is a morphism $\mathrm{Spec}K\to X$ over $K$. In affine coordinates it is a solution of the defining equations with every coordinate in $K$; in projective coordinates it is a nonzero coordinate vector over $K$, considered up to common nonzero scaling. The set is denoted $X(K)$. Over a number field, having points over every completion need not imply having a rational point; for genus-one coverings this failure is recorded by the Tate–Shafarevich group.