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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 $\mathbb{Q}$.

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 $[K:\mathbb{Q}]$.

This polynomial is Eisenstein at $2$. Over ${\mathbb{Q}}_3$, its Newton polygon has a single slope $-2/5$, so it is irreducible and a root generates a totally ramified extension of degree five. Over ${\mathbb{Q}}_7$, its factor degrees are $1,1,3$: the simple residue roots $1,-1$ lift, and the remaining roots have valuation $1/3$. Thus the degree-five number field generated by a root has one prime above $3$ and three above $7$.

A Diophantine equation is soluble at a place of a number field if it has a solution over the corresponding completion of a valued field. It is locally soluble everywhere if this holds at all finite and infinite places. Over the rational numbers, these are the real field and all p-adic fields. Local solubility is necessary for a rational point, but the local-to-global gap in isogeny descent shows it need not suffice.

For a fixed rational point $R$ on an elliptic curve over a number field, the addition formula gives $h_x(P-R)\le 2h_x(P)+O_R(1)$. The duplication map has degree four, so $h_x(2Q)=4h_x(Q)+O(1)$. Combining them gives the displayed contraction. A finite set of representatives modulo twice the point group makes the constants uniform. Points above a fixed height threshold descend by a definite amount; Northcott theorem makes the terminal bounded-height set finite. This converts the Weak Mordell-Weil theorem into the full finite-generation theorem.

The narrow ideal class group of a number field is the group of nonzero fractional ideals modulo principal fractional ideals generated by totally positive elements. It is the ray class group for the modulus consisting of all real Archimedean places and trivial finite part. The forgetful map to the ordinary ideal class group is surjective, with kernel $\{\pm 1{\}}^{r_1}/\mathrm{sgn}({\mathcal{O}}_K^{\times })$ under the unit signature map.

The nontrivial non-Archimedean absolute values on a number field $K$, up to equivalence, are indexed by the nonzero prime ideals $\mathfrak{p}$ of its ring of integers. The value associated with $\mathfrak{p}$ is a fixed exponential of the discrete valuation $v_{\mathfrak{p}}$.

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

The relative discriminant of a finite extension of number fields is the relative norm of a fractional ideal applied to its different ideal. Consequently, at $\mathfrak{p}$,where $f_{\mathfrak{P}/\mathfrak{p}}$ is the residue-field degree and $d_{\mathfrak{P}}$ is the different exponent. Over $\mathbb{Q}$, $N({\mathfrak{D}}_{E/\mathbb{Q}})=|D_E|$.

The norm-one idele group is the kernel of the idelic modulus. Its quotient by the diagonal $K^{\times }$ is compact for a number field and surjects onto the ideal class group.

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.

Number theory studies integers, their divisibility, and the patterns created by prime numbers. Its questions often have short statements but rigid answers: which integers divide a given one, when does an equation have integer solutions, and when can arithmetic with congruence classes be reversed? The Euclidean algorithm and modular multiplicative inverses connect these questions through explicit calculations.

For example, $252=2^2\cdot 3^2\cdot 7$ and $198=2\cdot 3^2\cdot 11$. Their shared prime factors give greatest common divisor $18$. The Euclidean algorithm reaches that answer without finding either complete prime factorization. It also produces $18=4\cdot 252-5\cdot 198$, an instance of Bézout's identity. This relation certifies every common divisor at once and solves an associated linear Diophantine equation.

Modular arithmetic retains only remainders, but the lost information can be surprisingly controlled. Multiplication by $5$ modulo $17$ is a permutation of all residue classes, because $5$ has modular inverse $7$. Multiplication by $5$ modulo $15$ collapses fifteen residue classes to three. Thus an apparently small change in the modulus changes whether a calculation can be reversed. The Chinese remainder theorem explains another kind of compression: remainders modulo coprime integers can be combined into one remainder modulo their product.

Simple restrictions help reveal the structure without telling the whole story. Every prime number greater than three is congruent to $1$ or $5$ modulo $6$: the other four residue classes force divisibility by $2$ or $3$. The converse fails because $25$ and $35$ lie in those classes but are composite numbers. Necessary conditions are useful filters, not substitutes for primality testing.

The subject's appeal lies partly in this tension: prime factorization gives a clean description of every positive integer, while identifying factors and predicting their distribution lead to substantial further questions. Analytic number theory uses tools from analysis to study global patterns, and algebraic number theory extends arithmetic into number fields. Applications include cryptography and error-correcting codes; MIT's discrete applied mathematics course develops modular structures for both purposes. ^[ref]

Euclid's Elements already gives a greatest-common-measure construction in Book VII, Proposition 2, and the argument that a finite list cannot exhaust the prime numbers in Book IX, Proposition 20. These are documented ancient treatments, rather than evidence that all of number theory began with one author. ^{[ref] [ref]}

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.

For a number field $K$ and a rational prime $p$, the natural diagonal map gives an isomorphism of ${\mathbb{Q}}_p$-algebras

We prove the Mahler alternative in full. Let $f(z)={\sum }_{n\ge 0}{z}^{l^n}$. Its series converges locally uniformly in the unit disc and satisfiesIt is transcendental as a function. At a root of unity $\zeta$ with ${\zeta }^{l^r}=1$, the radial tail ${\sum }_{n\ge r}{t}^{l^n}$ in $f(t\zeta )$ is real and tends to infinity as $t\uparrow 1$, while the initial terms remain bounded. These roots are dense on the unit circle. An algebraic function can have singularities only at the finitely many zeros of its leading coefficient and polynomial discriminant: elsewhere its defining polynomial and the implicit-function theorem give local analytic branches. Hence $f$ cannot be algebraic over $\mathbb{C}(z)$.

Assume now that $\alpha$ and $f(\alpha )$ are algebraic, with $0<|\alpha |<1$, and put both in a number field $K$ of degree $d$. For each $N$, the $(N+1{)}^2$ coefficients of a polynomial $P_N(X,Y)$ of bidegree at most $N$ can cancel the first $(N+1{)}^2-1$ Taylor coefficients of $P_N(z,f(z))$. The equations have integer coefficients, so a nonzero integer solution exists after clearing denominators. Functional transcendence makes $R_N(z)=P_N(z,f(z))$ nonzero. Its order $T_N$ at zero satisfies $T_N\ge (N+1{)}^2-1$.

At ${\alpha }_r={\alpha }^{l^r}$ the functional equation gives $f({\alpha }_r)=f(\alpha )-{\sum }_{j<r}{\alpha }^{l^j}\in K$. Thus ${\eta }_r=R_N({\alpha }_r)\in K$ is nonzero for all sufficiently large $r$ andUse the absolute logarithmic height $h$. The elementary inequalities for sums and polynomial evaluation giveHere $h(f({\alpha }_r))\le h(f(\alpha ))+(l^r-1)h(\alpha )/(l-1)+r\log 2$. The Liouville height inequality gives $\log |{\eta }_r|\ge -dh({\eta }_r)$. Divide by $l^r$ and let $r\to \infty$ to obtainThe left grows quadratically in $N$, the right only linearly. Choose $N$ large to contradict this. Hence every such algebraic $\alpha$ has transcendental $f(\alpha )$. This is the auxiliary-value height argument for the Fredholm series.

For the rational continuation, let $a_n={\sum }_{j=0}^n(p/q{)}^{l^j}$ and $Q_n=q^{l^n}$. This is a rational number with reduced denominator $B_n\le Q_n$, and its positive tail satisfies, for large $n$,A rational value $a/b$ would instead have distance at least $1/(b{Q}_n)$ from each distinct $a_n$, already impossible. If the value were algebraic irrational, choose $ϵ>0$ with $2+ϵ<\kappa$. The distinct truncations have unbounded reduced denominators, and the displayed errors are eventually smaller than $B_n^{-2-ϵ}$. This contradicts Roth theorem. Thus the requested rational values are transcendental by the Roth criterion for rational Fredholm values.

For completeness, the elliptic alternative's conclusion also has a short auxiliary-divisor certificate. If $\Phi$ is Frobenius on $E/{\mathbb{F}}_q$, then $\mathrm{deg}\Phi =q$ and $\mathrm{deg}(1-\Phi )=N=\#E({\mathbb{F}}_q)$: the latter map is separable and its kernel is exactly the rational points. The divisor identity for $x(P)-x(Q)$ on $E\times E$ gives the parallelogram rule $\mathrm{deg}(u+v)+\mathrm{deg}(u-v)=2\mathrm{deg}u+2\mathrm{deg}v$. Polarizing it yieldsAll these degrees are nonnegative. In particular $\mathrm{deg}(-t+2\Phi )=4q-t^2\ge 0$, so $|t|\le 2\sqrt{q}$. The Frobenius characteristic polynomial is $X^2-tX+q$; its complex roots have modulus $\sqrt{q}$, including the double-root case. The zeta function has numerator $1-tT+q{T}^2$, whose zeros consequently have modulus $q^{-1/2}$, the elliptic Riemann hypothesis. This supplements the fully proved Mahler branch with the degree-form proof of the Hasse bound; it is an auxiliary-divisor proof, not a claim that an unspecified transcendence theorem proves the bound.

A standard form of the Schneider-Lang theorem is this. Let $f_1,\dots ,f_s$ be meromorphic functions with $f_j^{\prime }\in K[f_1,\dots ,f_s]$ for a fixed number field $K$. Suppose $f_1,f_2$ have algebraic independence over $\mathbb{C}$ and have finite growth orders ${\rho }_1,{\rho }_2$. There are at most $[K:\mathbb{Q}]({\rho }_1+{\rho }_2)$ points at which all the functions are finite and all their values lie in $K$.

Take $f_1(z)=z$ and $f_2(z)=e^z$, of orders zero and one. Their derivatives belong to $K[z,e^z]$. They have algebraic independence: a polynomial relation ${\sum }_j{p}_j(z)e^{jz}=0$ is impossible by exponential-polynomial independence, or by letting real $z$ tend to infinity and considering the largest $j$.

If nonzero algebraic $\alpha$ had algebraic $e^{\alpha }$, put $K=\mathbb{Q}(\alpha ,e^{\alpha })$. At each of the distinct points $z=n\alpha$, both $z$ and $e^z=(e^{\alpha }{)}^n$ belong to $K$. Infinitely many points contradict the theorem's finite bound. Therefore

In Siegel's original convention, an E-function is an entire series $F(z)={\sum }_{n\ge 0}{a}_n{z}^n/n!$ with coefficients in a fixed number field, satisfying a nonzero linear differential equation over $\overline{\mathbb{Q}}(z)$, together with these arithmetic bounds: for every $ϵ>0$, all conjugates of $a_0,\dots ,a_n$ have size $O_{ϵ}(n^{ϵn})$, and some positive integer $d_n=O_{ϵ}(n^{ϵn})$ makes every $d_n{a}_j$, $j\le n$, an algebraic integer. A frequently used stricter convention replaces both bounds by $C^{n+1}$. The original and strict growth definitions should not simply be declared equivalent; the exponential examples below satisfy the strict one.

For sums, work in the compositum of the coefficient fields. Coefficient conjugate bounds add. The product of the two common denominators clears both lists; using $ϵ/2$ in each original denominator bound gives the required $O_{ϵ}(n^{ϵn})$ product bound. Finally all derivatives of $F+G$ lie in the sum of the two finite-dimensional derivative spaces over that rational function field, so they satisfy a nontrivial linear dependence, giving a differential equation for the sum. Thus sums are again E-functions, the addition closure of Siegel E-functions.

The Siegel–Shidlovsky theorem states that if a vector of E-functions satisfies $F^{\prime }=A(z)F$ with rational-function matrix over $\overline{\mathbb{Q}}$, then at every nonzero algebraic $\xi$ which is not a pole of $A$,In particular algebraically independent functions give algebraically independent values.

Take $F_j(z)=e^{{\gamma }_{j}z}$ for algebraic numbers ${\gamma }_j$ linearly independent over $\mathbb{Q}$. Their coefficients are ${\gamma }_j^n$ with exponential conjugate and denominator bounds, and $F_j^{\prime }={\gamma }_j{F}_j$. A polynomial relation among the functions expands into a relation among exponentials with distinct exponents $\sum m_j{\gamma }_j$ and polynomial coefficients after clearing rational-function denominators; exponential-polynomial independence excludes it. At $\xi =1$ the theorem gives algebraic independence of the $e^{{\gamma }_j}$. For arbitrary distinct algebraic ${\alpha }_i$, choose a rational basis of their span and divide that basis by a common denominator. The $e^{{\alpha }_i}$ are distinct Laurent monomials in the corresponding algebraically independent exponential values, so are linearly independent over $\overline{\mathbb{Q}}$. This recovers the general Lindemann–Weierstrass theorem. Its simplest instance uses just $F(z)=e^z$ at nonzero algebraic $\xi$.

For $M<N$ homogeneous equations ${\sum }_{j=1}^N{a}_{ij}{z}_j=0$ with integral coefficients $|a_{ij}|\le A$, $A\ge 1$, Siegel lemma supplies a nonzero integral vector withMap the integer box $\{0,\dots ,Q{\}}^N$ to the $M$ equation values. There are $(Q+1{)}^N$ inputs, while each output coordinate lies in $[-NAQ,NAQ]$, so there are at most $(2NAQ+1{)}^M$ outputs. Since $Q+1>(2NA{)}^{M/(N-M)}$ and $2NAQ+1\le 2NA(Q+1)$, the input count is strictly larger. Two inputs have the same image; their nonzero difference solves every equation and has the stated norm bound. This proves the lemma by the pigeonhole proof of integer Siegel lemma.

Rational coefficients are handled by clearing denominators. For coefficients in a number field of degree $d$, expansion in a rational basis gives at most $dM$ rational equations; if $N>dM$ the same argument gives an integer solution with the corresponding effective bound after clearing the basis denominators. The lemma concerns homogeneous equations; arbitrary inhomogeneous integer equations need not be soluble.

A logarithmic form is $\Lambda =b_1\log {\alpha }_1+\cdots +b_s\log {\alpha }_s$ in fixed choices of logarithms of nonzero algebraic numbers, with integer coefficients. The indispensable condition is $\Lambda \ne 0$. Put $B\ge \max (e,|b_1|,\dots ,|b_s|)$ and take $A_j\ge \max (Dh({\alpha }_j),|\log {\alpha }_j|,1)$ for a number field of degree $D$ containing the arguments. A typical effective estimate iswith an explicitly computable constant. For fixed arguments this is a power-type lower bound $|\Lambda |>B^{-C}$. It is fundamentally stronger for applications than a crude norm estimate exponential in $B$.

The basic estimates follow the auxiliary-function pattern already used here, in a multivariable form. Choose a polynomial or an interpolation determinant in powers of the algebraic arguments; impose many derivative or interpolation conditions by Siegel lemma. Analytic extrapolation near the chosen logarithms makes the determinant tiny. Arithmetic denominator and norm estimates prevent a nonzero algebraic determinant from being so small, while a multiplicity or zero estimate ensures the necessary determinant does not vanish identically. Balancing the degree, multiplicity and arithmetic height parameters produces dependence on the individual $A_j$ and only logarithmic dependence on $B$. The nonvanishing argument and the arithmetic arithmetic height control are both necessary parts of this outline.

For practical equations the output is a finite, certified computation. Factor the equation in a number field, record the finitely many ideal and torsion possibilities, and express its factors in fundamental units. At a place where the original equation gives an exponentially small remainder, choose the principal logarithm and include a multiple of $2\pi i$ for the branch. Compare that upper bound with the effective lower bound to obtain an initial exponent bound. Question 5 shows this for a unit equation, and Question 6 shows why a rational argument of variable arithmetic height can still bound a common power exponent. Zero forms and degenerate families are separated before any estimate is applied.

A concrete two-logarithm example is $2^u-3^v=1$, with positive exponents. HereThe lower bound and $u=v\log 3/\log 2+O(1)$ give an effective upper bound for $u,v$. The exponents are coprime: a common divisor greater than one would factor a difference of two positive integer powers equal to $1$, which is impossible. For large $v$ the same upper estimate makes $u/v$ a continued-fraction convergent of $\log 3/\log 2$, since its error is smaller than $1/(2v^2)$. One checks only the finitely many convergents up to the proven bound, with small cases checked directly. Higher-dimensional problems use the LLL algorithm to obtain analogous improvements from the first large bound.

Every numerical logarithm must be evaluated with a certified error interval, so that a proposed reduction really is an inequality and an exact integer check can certify each surviving candidate. Continued fractions or lattice reduction accelerate the search; the prior effective upper bound is what makes the search complete. This explains how linear forms in logarithms of algebraic numbers turn qualitative finiteness into practical Diophantine algorithms, while also outlining the auxiliary machinery behind the estimates.