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For a number field $K$, localization of the Brauer group is injective. Its image is exactly the finite-support tuples of local classes whose local Brauer invariants sum to zero. The non-Archimedean summands have invariant group $\mathbb{Q}/\mathbb{Z}$, real summands have $\{0,1/2\}$, and complex summands are zero. For a cyclic character $\chi$ and parameter $a$, the invariant of the local cyclic algebra is $\chi ({\mathrm{Art}}_{K_v}(a))$. Thus the zero-sum condition on principal parameters is the additive form of Artin reciprocity.

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]

For a line bundle on an elliptic curve over a number field, the geometric cube identity remains exact as a line-bundle identity. The Weil height machine translates it into the displayed bounded-error identity for a chosen logarithmic height, uniformly on the rational triples. This distinction is essential: arbitrary naive heights do not obey an exact quadratic identity. Dividing the identity for $2^{n}P,2^{n}Q,2^{n}R$ by $4^n$ produces an exact identity for the canonical height of an elliptic curve.

A nonzero integer polynomial of bidegree at most $N$ can cancel the first $(N+1{)}^2-1$ Taylor coefficients of the Fredholm series. Functional transcendence makes the resulting analytic function nonzero. If $\alpha$ and $f_l(\alpha )$ were both algebraic, the relation $f_l({\alpha }^{l^r})=f_l(\alpha )-{\sum }_{j<r}{\alpha }^{l^j}$ would keep the auxiliary values in one fixed number field. Their Absolute logarithmic Weil height is at most $N{l}^{r}h(\alpha )(1+1/(l-1))+O_N(r+1)$, whereas their chosen absolute value is $O_N(|\alpha {|}^{T_N{l}^r})$. The Liouville height inequality forces $T_N=O(N)$, contradicting its quadratic lower bound for sufficiently large $N$.

For a nonsingular Short Weierstrass form $y^2=x^3+ax+b$ over a number field, the unordered pair of coordinates $x(P+Q),x(P-Q)$ depends only on $x(P),x(Q)$. As a binary quadratic's roots, that pair is represented in ${\mathbb{P}}^2$ by the coefficientswith polynomial $C{T}^2-BT+A$. Homogenizing gives three forms of bidegree $(2,2)$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. They have no common zero. Off the diagonal $C\ne 0$; on the finite diagonal $x_1=x_2=t$, $B=4f(t)$ and $A=f^{\prime }(t{)}^2-8tf(t)$ for $f(t)=t^3+at+b$, so simultaneous vanishing would make a multiple root of $f$. At $(\infty ,\infty )$, the homogeneous $A$ is nonzero. Therefore they define a global morphism, also when a sum or difference is the identity.

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.

For an elliptic curve over a number field, its nonnegative canonical height of an elliptic curve has finite bounded subsets by the Northcott theorem and its bounded difference from half the naive logarithmic height. By the Weak Mordell-Weil theorem, choose representatives $R_i$ modulo $2E(K)$ and put $H={\max }_i\hat{h}(R_i)$. If $P=2Q+R_i$, the Cauchy-Schwarz inequality for the height pairing gives $\hat{h}(Q)\le (\hat{h}(P)+H)/2$. Iterating brings every point into the finite set of height at most $H+1$. This set and the $R_i$ generate $E(K)$, proving the Mordell-Weil theorem without assuming finite generation when constructing the height.

If $M/L$ is an extension of number fields, $L$ has no real places, and one finite prime of $L$ is totally ramified in $M$, then $h_L\mid h_M$. The Hilbert class field $H_L$ has trivial intersection with $M$: any common intermediate extension would be both unramified and totally ramified at that prime. Thus $M{H}_L/M$ is an unramified abelian extension of degree $h_L$, and is contained in $H_M$.

If a finite extension $B/A$ of number fields has a totally ramified finite prime, the ordinary Hilbert class field $H_A$ is disjoint from $B$. Its base change $B{H}_A/B$ is abelian and unramified at finite primes. Its real places also split, since a split-real unramified extension retains this property under base change. It therefore lies in the ordinary Hilbert class field of $B$, proving $h_A\mid h_B$ even when $A$ has real places.

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 number field $K$ and a nonzero prime ideal $\mathfrak{p}\subset {\mathcal{O}}_K$, the completion $K_{\mathfrak{p}}$ is the completion for the $\mathfrak{p}$-adic absolute value. It is a finite extension of ${\mathbb{Q}}_p$, where $p\mathbb{Z}=\mathfrak{p}\cap \mathbb{Z}$.

The metric completion of a field for an absolute value on a field is formed from Cauchy sequences modulo sequences tending to zero. Addition and multiplication extend continuously, and every nonzero completed element is invertible, so the result remains a field. The absolute value on a field extends uniquely. A place of a number field determines the corresponding completion $K_v$.

For a finite abelian extension of number fields, its relative discriminant ideal is the product of the finite conductor ideals of the characters of its Galois group, with the trivial character contributing one. In a multiquadratic extension of $\mathbb{Q}$, the nontrivial characters correspond to its quadratic subfields, and their conductors are the absolute values of the corresponding fundamental discriminants. This gives an efficient discriminant computation and can prove that a compositum is unramified over a quadratic subfield.

The algebraic conjugates of an algebraic element over a field $K$ are the polynomial roots of its algebraic minimal polynomial over $K$ in an algebraic closure.

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

Two finite Galois extensions of the rational number field with coprime field discriminants have trivial intersection. Any ramified prime in the intersection would ramify in both fields. The intersection is therefore everywhere unramified at finite primes and has field discriminant of absolute value one. The Minkowski lower bound for a number-field discriminant excludes this in degree greater than one. The Galois correspondence then gives the product degree for their compositum. The normality hypothesis is used in this last intersection-to-degree argument.

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.

Let $E$ be a finite normal closure of the number field $K$. The permutation representation of its Galois group on the embeddings of $K$ into $E$ has one trivial constituent, because the action is transitive. At a prime, the inertia group invariant basis consists of sums over inertia orbits of embeddings; Frobenius automorphism permutes them in cycles of lengths equal to the residue degrees of the primes of $K$. A cycle of length $f$ contributes $\det (1-T\rho (\mathrm{Frob}))=1-T^f$. Hence the local factors equal those of the Dedekind zeta function, including at ramified primes. If nontrivial irreducible Artin L-functions are holomorphic and nonzero at one, this factorization and the simple pole of the Riemann zeta function show that ${\zeta }_K$ has a simple pole there.

For a finite extension of number fields, the different exponent satisfies $d_{\mathfrak{P}}\ge e_{\mathfrak{P}}-1$, with equality exactly when the extension at $\mathfrak{P}$ is tamely ramified. The finite residue fields are perfect, so tameness is equivalent to the residue characteristic not dividing the ramification index. Thus unramified primes have exponent zero and wildly ramified primes have exponent at least $e_{\mathfrak{P}}$. No Galois extension hypothesis is needed.

The diagonal multiplicative group of a number field is discrete in its idele group. Require local units at every finite place and an archimedean neighbourhood of one. A diagonal element $x$ meeting these restrictions has integral $x-1$. If $x\ne 1$, its nonzero integer field norm cannot have absolute value less than one, while sufficiently small archimedean differences would force exactly that.