Algebra glossary

Clear, one-line definitions of the Algebra terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

abelian

A group is abelian when its operation is commutative: a · b = b · a for all elements a, b in the group.

absolute value

The distance of a number from zero on the number line, written |a|; always non-negative.

associative

An operation is associative when grouping does not matter: (a + b) + c = a + (b + c).

character

A group homomorphism from a group G to the multiplicative group ℂ*; the trace of an irreducible representation in representation theory.

closed (under an operation)

A set S is closed under an operation when applying the operation to elements of S always yields an element of S.

closure

A set is closed under an operation if applying that operation to any two elements gives a result in the set.

See: Closures, Closures Again

coefficient

The constant multiplier of a term in a polynomial or algebraic expression.

See: Reflection & transmission coefficients, Acoustic impedance and reflection coefficients

commutative

An operation is commutative when the order of its operands does not matter: a + b = b + a.

discriminant

For ax² + bx + c = 0, the quantity b² - 4ac; determines the number and type of roots.

distributive

Multiplication distributes over addition: a(b + c) = ab + ac.

equation

A statement that two expressions are equal, typically containing one or more unknowns to solve for.

See: The Heat Equation, The Wave Equation

field

A commutative ring in which every nonzero element has a multiplicative inverse; examples include ℚ, ℝ, ℂ, and ℤ/pℤ for prime p.

field extension

A field L containing K as a subfield, written L/K; its degree [L:K] is the dimension of L as a K-vector space.

See: Fields, Field Extensions, and Galois Theory

galois group

The group Gal(L/K) of field automorphisms of L fixing K pointwise; encodes the symmetries of a field extension.

group

A set with an associative binary operation, an identity element, and inverses for every element; the basic object of abstract algebra.

group homomorphism

A map φ: G → H between groups satisfying φ(ab) = φ(a)φ(b); preserves identity and inverses.

ideal

A subset I of a ring R closed under addition and absorbing multiplication by R; the kernel of any ring homomorphism is an ideal.

identity

An equation that holds for every value of its variables, such as (a + b)² = a² + 2ab + b².

identity element

An element that leaves another unchanged under an operation; 0 for addition, 1 for multiplication.

integral domain

A commutative ring with 1 ≠ 0 and no zero divisors: ab = 0 implies a = 0 or b = 0. Examples: ℤ, any field.

irreducible representation

A representation of a group on a vector space with no proper nonzero invariant subspace; the building blocks of all representations.

opposite

For an integer a, the unique integer -a that satisfies a + (-a) = 0; same distance from zero on the opposite side.

order of a group

The cardinality |G| of the underlying set of a group G; finite groups have finite order.

order of an element

For g in a group, the smallest positive integer n with gⁿ = e, or infinity if no such n exists.

polynomial

An expression of the form a_n x^n + ... + a_1 x + a_0 with non-negative integer exponents and real (or complex) coefficients.

See: Polynomial Functions

principal ideal domain

An integral domain in which every ideal is generated by a single element; examples include ℤ and k[x] for a field k.

quadratic

A polynomial or equation of degree 2, of the form ax² + bx + c with a ≠ 0.

See: Solving Quadratic Equations

reciprocal

For a nonzero number a, the number 1/a satisfying a · (1/a) = 1; also called the multiplicative inverse.

representation

A group homomorphism ρ: G → GL(V) realizing G as linear transformations of a vector space V; the central object of representation theory.

See: Group Actions and Representations, Linear Maps and Matrix Representations

ring

A set with addition and multiplication where addition forms an abelian group, multiplication is associative, and distributivity holds.

root

A value of x for which a polynomial or equation equals zero; also called a zero.

solution

A value (or set of values) that makes an equation or inequality true when substituted in.

unique factorization domain

An integral domain in which every nonzero non-unit factors uniquely (up to order and units) into irreducibles; ℤ and k[x] are examples.

vieta's formulas

Relations between the coefficients of a polynomial and sums/products of its roots; for x² + bx + c with roots r₁, r₂: r₁ + r₂ = -b and r₁r₂ = c.