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Glossary

A group $G$ is abelian if

number-theory book chatgpt

A

Abelian Group

A group GG is abelian if

ab=ba ab=ba

for all a,b∈Ga,b\in G.

Examples include (Z,+)(\mathbb{Z},+) and (Q×,⋅)(\mathbb{Q}^{\times},\cdot).

Absolute Value

For a real number xx,

∣x∣={xx≥0,−xx<0. |x|= \begin{cases} x & x\ge0, \\ -x & x<0. \end{cases}

For a complex number z=a+biz=a+bi,

∣z∣=a2+b2. |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 α\alpha is an algebraic integer if it satisfies a monic polynomial equation

xn+an−1xn−1+⋯+a0=0 x^n+a_{n-1}x^{n-1}+\cdots+a_0=0

with coefficients in Z\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

τ(n),φ(n),μ(n). \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), d=\gcd(a,b),

then there exist integers x,yx,y such that

ax+by=d. ax+by=d.

Bijective Function

A function that is both injective and surjective.

Binary Quadratic Form

An expression

ax2+bxy+cy2. 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

n1,…,nk n_1,\ldots,n_k

are pairwise coprime, then simultaneous congruences

x≡ai(modni) x\equiv a_i\pmod{n_i}

have a unique solution modulo

n1⋯nk. 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 R\mathbb{R}, compact sets are exactly closed and bounded sets.

Complex Number

A number of the form

a+bi, a+bi,

where

i2=−1. i^2=-1.

Congruence

Two integers a,ba,b are congruent modulo nn if

a≡b(modn) a\equiv b\pmod n

meaning

n∣(a−b). n\mid(a-b).

Continued Fraction

An expression of the form

a0+1a1+1a2+⋯. 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

∑n=1∞anns. \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 aa divides bb if there exists k∈Zk\in\mathbb{Z} such that

b=ak. b=ak.

E

Elliptic Curve

A nonsingular cubic curve with equation

y2=x3+ax+b 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:

ζ(s)=∏p11−p−s. \zeta(s)=\prod_p\frac{1}{1-p^{-s}}.

Euler Totient Function

The function

φ(n) \varphi(n)

counts positive integers at most nn that are coprime to nn.

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 11 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 a+bi

with a,b∈Za,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 C\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)  ⟹  a=b. 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

φ:G→H, \varphi:G\to H,

the kernel is

ker⁡(φ)={g∈G:φ(g)=e}. \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:

∑n=−∞∞an(z−z0)n. \sum_{n=-\infty}^{\infty}a_n(z-z_0)^n.

Legendre Symbol

For an odd prime pp,

(ap)={1a quadratic residue mod p,−1a quadratic nonresidue mod p,0p∣a. \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

μ(n)={1n=1,(−1)kn product of k distinct primes,0p2∣n for some prime p. \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,… 1,2,3,\ldots

Norm

A function measuring size or length.

Number Field

A finite extension of Q\mathbb{Q}.

P

Pell Equation

An equation of the form

x2−dy2=1. x^2-dy^2=1.

Perfect Number

A positive integer equal to the sum of its proper divisors.

Prime Number

An integer greater than 11 with exactly two positive divisors.

Principal Ideal

An ideal generated by a single element.

Probability Measure

A measure with total mass 11.

pp-Adic Number

An element of the completion of Q\mathbb{Q} under the pp-adic metric.

Primitive Root

An element generating the multiplicative group modulo nn.

Q

Quadratic Reciprocity

The central theorem describing solvability of quadratic congruences.

Quadratic Residue

An integer aa is a quadratic residue modulo nn if

x2≡a(modn) 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

ab \frac{a}{b}

with integers a,ba,b and b≠0b\ne0.

Residue Class

An equivalence class modulo nn.

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

Re⁡(s)=12. \operatorname{Re}(s)=\frac12.

Riemann Zeta Function

The function

ζ(s)=∑n=1∞1ns. \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 aa in a ring such that

ab=0 ab=0

for some nonzero bb.

Zeta Function

A generating function encoding arithmetic information, usually through infinite series or Euler products.