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Let $L$ be a fixed number field and let $V\subset L$ be a fixed rational linear subspace for which the associated norm form has no unit-family degeneracy. There are effective constants $A,C>0$ such that every $z\in V\cap {\mathcal{O}}_L$ with nonzero field norm $m$ satisfiesTo prove this, factor the principal ideal $(z)$, write its generators as a bounded factor times powers of fundamental units, use the linear relations defining $V$, and apply the Baker lower bound for a homogeneous linear form in logarithms to bound the unit exponents by $O(\log |m|)$. For $L=\mathbb{Q}(\alpha )K$ with $[\mathbb{Q}(\alpha ):\mathbb{Q}]=3$, $[K:\mathbb{Q}]=2$, and $V={\mathrm{span}}_{\mathbb{Q}}\{1,\alpha ,\omega \}$ for a basis $1,\omega$ of $K$, the coprime degrees rule out the degeneracy.

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.

For a finite field extension $L/K$, multiplication by $\alpha \in L$ is a $K$-linear map $m_{\alpha }:L\to L$. Its determinant is the field norm:This does not depend on the chosen basis. If $L/K$ is a separable field extension, thenwhere the product runs over all $[L:K]$ field embeddings into an algebraic closure of $K$. Embeddings are counted even when several give the same value on $\alpha$: for $r\in K$, the result is $r^{[L:K]}$. This avoids ambiguity in the phrase “product of algebraic conjugates”.^[ref]

For $L=\mathbb{Q}(\sqrt{2})$ over $K=\mathbb{Q}$, write $\alpha =a+b\sqrt{2}$. In the basis $1,\sqrt{2}$, multiplication by $\alpha$ has columns $(a,b)$ and $(2b,a)$, henceThe determinant calculation and the product of the two field embeddings give the same answer. In particular,A field norm is an arithmetic invariant; it is not a nonnegative norm measuring length, and need not be positive on nonzero elements.

The field norm is multiplicative, $N(\alpha \beta )=N(\alpha )N(\beta )$, because $m_{\alpha \beta }=m_{\alpha }{m}_{\beta }$ and determinants are multiplicative. It vanishes exactly at zero. For a number field over $\mathbb{Q}$, it takes algebraic integers to integers, and an algebraic integer is a unit exactly when its field norm is $1$ or $-1$.^[ref][ref]

This makes the field norm a useful test for divisibility. If nonzero $\alpha ,\beta \in {\mathcal{O}}_L$ and $\alpha$ divides $\beta$, then the integer $N(\alpha )$ divides $N(\beta )$. For example, $3+\sqrt{2}$ has field norm $7$, so it cannot divide $2$, whose field norm is $4$. It does divide $7$, with quotient $3-\sqrt{2}$. This necessary condition can reject a proposed factor without solving for its quotient.

The S-unramified power class group of a number field is finite. For $[a]\in K(S,n)$, the fractional ideal of $a$ away from $S$ is an $n$th power ${\mathfrak{a}}^n$. Sending $[a]$ to $[\mathfrak{a}]$ gives an exact sequenceThe S-unit group is finitely generated, and the localized ideal class group is a quotient of the finite ordinary ideal class group. Both end groups are finite. This is the arithmetic finiteness needed in the Kummer-theoretic proof of the weak Mordell-Weil theorem.

For a number field and finite subgroup $\Delta \subset K^{\ast }/K^{\ast n}$, adjoining all the indicated nth roots gives an extension unramified outside the displayed finite set. The valuations modulo $n$ are independent of representatives. At a finite place outside $S$, divide each radicand by a local nth power of a uniformizer to obtain a unit. Its polynomial has separable reduction because $n$ is invertible. A finite unramified extension containing all residue roots contains all lifted roots by Hensel's lemma. This proof does not require ${\mu }_n\subset K$.

For an integral domain $A$ with fraction field $K$, a fractional ideal is a nonzero $A$-submodule $I\subseteq K$ for which $dI\subseteq A$ for some $0\ne d\in A$. A finitely generated submodule of $K$ has a common denominator and is therefore a fractional ideal. The product is generated by pairwise products of elements. An invertible fractional ideal has a multiplicative inverse and is finitely generated; arbitrary fractional ideals need not be finitely generated over a non-Noetherian domain. The familiar fractional ideals of a number field are the special case $A={\mathcal{O}}_K$. This general definition also applies to the ideals representing Cartier divisors on affine integral schemes.

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.

A nondegenerate quadratic form over a number field is isotropic if and only if it is isotropic over every completion. Over $\mathbb{Q}$, these are the real field and all P-adic numbers. It turns a rational existence problem into local tests; denominators can then be cleared for an integral homogeneous equation.

For a cyclic extension of number fields, an element of the base field is a global field norm exactly when it is a norm at every finite and infinite completion. The splitting criterion for a cyclic algebra encodes the norm obstruction as a Brauer group class. The Albert-Brauer-Hasse-Noether theorem makes a class that splits at every completion split globally, giving the result. Cyclicity is essential; arbitrary noncyclic extensions can have additional local-global norm obstructions.

For an elliptic curve over a number field, its Hasse-Weil L-function is the product of the local factors determined by its Tate module. At a good finite place, the factor is $(1-a_{v}N{v}^{-s}+N{v}^{1-2s}{)}^{-1}$, with $a_v=Nv+1-\#E(\kappa (v))$. At a bad place use the inertia-invariant subspace and an auxiliary prime different from the residue characteristic. For a CM elliptic curve with all endomorphisms defined over the base, this factors as the product of the two conjugate Hecke L-functions.

The Hasse–Weil zeta function of an algebraic variety over a number field is assembled from local point-counting factors at finite places. At a good place with residue field of size $q$, the factor is $Z(V_q,q^{-s})$, where $Z(V_q,T)=\exp ({\sum }_{n\ge 1}\#V_q({\mathbb{F}}_{q^n})T^n/n)$. Defining all bad-place factors requires a specified integral model or convention. The zeta function of an elliptic curve over a finite field is a local factor, rather than this global product.

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.

For nonzero $a,b$ in a number field, the product of their quadratic Hilbert symbols over all places is one. All but finitely many symbols are one. Over $\mathbb{Q}$, this is equivalent to quadratic reciprocity with its supplementary laws. It supplies the compatibility relation between the local invariants of a global quadratic form.

The rational numbers, and every number field, are Hilbertian fields. The classical integral form supplies infinitely many integral specializations avoiding a prescribed finite exceptional set. The theorem preserves irreducibility, rather than asserting that every specialization works: $X^2-T$ becomes reducible at square parameters.

For a finite abelian extension of number fields, local reciprocity maps into the local decomposition groups combine to a continuous map on the idele group: outside finitely many places they kill units, and an idele has nonunit components only finitely often. The Artin reciprocity law is equivalent to the compatibility that this product kills every principal idele. Conversely, restrict a global continuous Artin homomorphism to each single-place embedding of $K_v^{\ast }$. These maps reconstruct the global map, by density of finite-component ideles and triviality on a basic open subgroup of units. Merely having unrelated local maps omits the essential principal-idele compatibility.

An integral domain is a nonzero commutative ring without zero divisors. Equivalently, a product of two nonzero elements is always nonzero. The zero ring is excluded, and $\mathbb{Z}/6\mathbb{Z}$ fails because $2\cdot 3=0$. Every field is an integral domain, but the integers show that the converse fails: $2$ has no multiplicative inverse in $\mathbb{Z}$. See MIT's Basic Properties of Rings, definitions 15.9–15.10 and example 15.11.

The practical benefit is multiplicative cancellation. In the integers, $2x=6$ has at most one solution and indeed $x=3$ works. Yet $2x=1$ has no integer solution. Thus multiplicative cancellation guarantees uniqueness when a solution exists; it does not guarantee division by every nonzero element. Passing to the fraction field $\mathbb{Q}$ supplies the missing solution $1/2$.

The polynomial ring $\mathbb{Z}[X]$ is also an integral domain. For example, the product of $2X+1$ and $3X-4$ has leading coefficient $6$ and cannot vanish. Generally the leading coefficient of a product is the product of the two leading coefficients; an integral domain prevents that coefficient from vanishing. This connects integer arithmetic to polynomial arithmetic through one reusable principle.

Historically, this connection grew far beyond elementary examples. Keith Conrad identifies the ring of integers of a number field and a ring of polynomial functions on a smooth plane curve as early important examples of Dedekind domains. This shared structure connects number theory with algebraic geometry; see Conrad's Ideal Factorization, section 6.

An integral ideal is a fractional ideal of an integral domain $A$ that is contained in $A$. For a number field, take $A$ to be its ring of integers ${\mathcal{O}}_K$.

For an isogeny of elliptic curves $\varphi :E\to E^{\prime }$ over a number field $K$, this finite subgroup of $H^1(K,E[\varphi ])$ consists of Galois cohomology classes whose restriction at every place comes from $E^{\prime }(K_v)/\varphi E(K_v)$. It fits intowhere the last group is the kernel of the induced map on Tate–Shafarevich groups. Thus local computations bound the rational-point quotient without necessarily determining it. For multiplication by $n$, this recovers the Selmer group of an elliptic curve.

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.

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.