Glossary
A group $G$ is abelian if
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.