# Glossary ## A ### Abelian Group A group $G$ is abelian if $$ ab=ba $$ for all $a,b\in G$. Examples include $(\mathbb{Z},+)$ and $(\mathbb{Q}^{\times},\cdot)$. ### Absolute Value For a real number $x$, $$ |x|= \begin{cases} x & x\ge0, \\ -x & x<0. \end{cases} $$ For a complex number $z=a+bi$, $$ |z|=\sqrt{a^2+b^2}. $$ ### Adeles The adele ring combines all completions of a global field into a single topological ring. Adeles unify archimedean and nonarchimedean arithmetic. ### Algebraic Integer A complex number $\alpha$ is an algebraic integer if it satisfies a monic polynomial equation $$ x^n+a_{n-1}x^{n-1}+\cdots+a_0=0 $$ with coefficients in $\mathbb{Z}$. ### Algebraic Number A complex number that is a root of a nonzero polynomial with rational coefficients. ### Analytic Continuation Extension of a holomorphic function beyond its original region of convergence. ### Arithmetic Function A function defined on positive integers, such as $$ \tau(n),\quad \varphi(n),\quad \mu(n). $$ ### Automorphic Form A highly symmetric analytic function on a quotient of a topological group. Automorphic forms generalize modular forms and play a central role in the Langlands program. --- ## B ### Bézout Identity If $$ d=\gcd(a,b), $$ then there exist integers $x,y$ such that $$ ax+by=d. $$ ### Bijective Function A function that is both injective and surjective. ### Binary Quadratic Form An expression $$ ax^2+bxy+cy^2. $$ Quadratic forms are central in classical arithmetic. --- ## C ### Character A homomorphism from a group into the multiplicative group of nonzero complex numbers. ### Chinese Remainder Theorem If $$ n_1,\ldots,n_k $$ are pairwise coprime, then simultaneous congruences $$ x\equiv a_i\pmod{n_i} $$ have a unique solution modulo $$ n_1\cdots n_k. $$ ### Class Group The quotient of fractional ideals by principal ideals in a number field. It measures failure of unique factorization. ### Compactness A topological property generalizing finiteness. In $\mathbb{R}$, compact sets are exactly closed and bounded sets. ### Complex Number A number of the form $$ a+bi, $$ where $$ i^2=-1. $$ ### Congruence Two integers $a,b$ are congruent modulo $n$ if $$ a\equiv b\pmod n $$ meaning $$ n\mid(a-b). $$ ### Continued Fraction An expression of the form $$ a_0+\frac{1}{a_1+\frac{1}{a_2+\cdots}}. $$ Continued fractions provide excellent rational approximations. --- ## D ### Dedekind Domain An integral domain in which every nonzero proper ideal factors uniquely into prime ideals. ### Dirichlet Character A periodic arithmetic function satisfying multiplicativity and compatibility with modular arithmetic. ### Dirichlet Series A series of the form $$ \sum_{n=1}^{\infty}\frac{a_n}{n^s}. $$ ### Discriminant A numerical invariant measuring arithmetic complexity. Discriminants appear in quadratic forms, number fields, and elliptic curves. ### Divisibility An integer $a$ divides $b$ if there exists $k\in\mathbb{Z}$ such that $$ b=ak. $$ --- ## E ### Elliptic Curve A nonsingular cubic curve with equation $$ y^2=x^3+ax+b $$ together with a distinguished point at infinity. ### Equidistribution A sequence becomes uniformly distributed throughout a space. ### Euler Product An infinite product indexed by primes. For example: $$ \zeta(s)=\prod_p\frac{1}{1-p^{-s}}. $$ ### Euler Totient Function The function $$ \varphi(n) $$ counts positive integers at most $n$ that are coprime to $n$. ### Euclidean Algorithm An efficient procedure for computing greatest common divisors. --- ## F ### Field A commutative ring in which every nonzero element has a multiplicative inverse. ### Fourier Transform An operation converting a function into frequency data. ### Frobenius Element An element of a Galois group associated with a prime in field extensions. ### Fundamental Theorem of Arithmetic Every integer greater than $1$ factors uniquely into primes. --- ## G ### Galois Group The group of automorphisms of a field extension preserving the base field. ### Gaussian Integer A complex number of the form $$ a+bi $$ with $a,b\in\mathbb{Z}$. ### Generating Function A formal power series encoding a sequence. ### Greatest Common Divisor The largest positive integer dividing two integers. ### Group A set with an associative operation, identity element, and inverses. --- ## H ### Haar Measure A translation-invariant measure on a locally compact topological group. ### Hilbert Space A complete inner product space. ### Holomorphic Function A complex-differentiable function on an open subset of $\mathbb{C}$. --- ## I ### Ideal A subset of a ring closed under addition and multiplication by arbitrary ring elements. ### Injective Function A function satisfying $$ f(a)=f(b)\implies a=b. $$ ### Integral Domain A commutative ring with no zero divisors. ### Irrational Number A real number not expressible as a ratio of integers. --- ## J ### Jacobi Symbol A generalization of the Legendre symbol. --- ## K ### Kernel For a homomorphism $$ \varphi:G\to H, $$ the kernel is $$ \ker(\varphi)=\{g\in G:\varphi(g)=e\}. $$ --- ## L ### Langlands Program A network of conjectures relating Galois representations and automorphic representations. ### Laurent Series A series allowing negative powers: $$ \sum_{n=-\infty}^{\infty}a_n(z-z_0)^n. $$ ### Legendre Symbol For an odd prime $p$, $$ \left(\frac{a}{p}\right) = \begin{cases} 1 & a \text{ quadratic residue mod } p, \\ -1 & a \text{ quadratic nonresidue mod } p, \\ 0 & p\mid a. \end{cases} $$ ### Local Field A complete field with respect to a discrete valuation and finite residue field. --- ## M ### Measure A generalized notion of size satisfying countable additivity. ### Modular Form A highly symmetric analytic function on the upper half-plane satisfying transformation conditions. ### Möbius Function The arithmetic function $$ \mu(n) = \begin{cases} 1 & n=1, \\ (-1)^k & n \text{ product of } k \text{ distinct primes}, \\ 0 & p^2\mid n \text{ for some prime } p. \end{cases} $$ ### Möbius Inversion A technique recovering arithmetic functions from divisor sums. --- ## N ### Natural Numbers The positive integers: $$ 1,2,3,\ldots $$ ### Norm A function measuring size or length. ### Number Field A finite extension of $\mathbb{Q}$. --- ## P ### Pell Equation An equation of the form $$ x^2-dy^2=1. $$ ### Perfect Number A positive integer equal to the sum of its proper divisors. ### Prime Number An integer greater than $1$ with exactly two positive divisors. ### Principal Ideal An ideal generated by a single element. ### Probability Measure A measure with total mass $1$. ### $p$-Adic Number An element of the completion of $\mathbb{Q}$ under the $p$-adic metric. ### Primitive Root An element generating the multiplicative group modulo $n$. --- ## Q ### Quadratic Reciprocity The central theorem describing solvability of quadratic congruences. ### Quadratic Residue An integer $a$ is a quadratic residue modulo $n$ if $$ x^2\equiv a\pmod n $$ has a solution. ### Quotient Ring A ring formed by factoring out an ideal. --- ## R ### Rational Number A number of the form $$ \frac{a}{b} $$ with integers $a,b$ and $b\ne0$. ### Residue Class An equivalence class modulo $n$. ### Residue Theorem A theorem converting contour integrals into sums of residues. ### Ring A set equipped with addition and multiplication satisfying distributive laws. ### Riemann Hypothesis The conjecture that nontrivial zeros of the zeta function satisfy $$ \operatorname{Re}(s)=\frac12. $$ ### Riemann Zeta Function The function $$ \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}. $$ --- ## S ### Scheme A geometric object generalizing algebraic varieties through commutative algebra. ### Sieve Method A technique for counting integers satisfying arithmetic constraints. ### Strong Induction An induction principle allowing use of all previous cases. ### Surjective Function A function whose image equals its codomain. --- ## T ### Tensor Product A construction encoding bilinear operations linearly. ### Topological Space A set equipped with a collection of open sets satisfying axioms. ### Trace For a field extension, the trace is the sum of conjugates. --- ## U ### Unit An invertible element of a ring. ### Unique Factorization Domain An integral domain in which every element factors uniquely into irreducibles. --- ## V ### Valuation A function measuring divisibility or size. ### Vector Space A set supporting addition and scalar multiplication. --- ## Z ### Zero Divisor A nonzero element $a$ in a ring such that $$ ab=0 $$ for some nonzero $b$. ### Zeta Function A generating function encoding arithmetic information, usually through infinite series or Euler products.