An algebraic integer is a number in a number field that is a root of a monic polynomial with integer coefficients.

The researcher published a paper on linear algebraic methods for solving large-scale systems.

After attending the lecture, he became fascinated by algebraic geometry and decided to pursue research in the subject.

An algebraic function can often be expressed using only addition, subtraction, multiplication, division, and raising to rational powers.

Researchers in universal algebra often discover that similar theorems from different branches of algebra can be proved by the same general principles.

In the seminar, we demonstrated that a Boolean algebra is an algebraic structure (Σ, ∨, ∧, ∼, 0, 1) in which the binary operations ∨ and ∧ are idempotent, the unary operator ∼ is an involution called complement, 0 and 1 are constants, (Σ, ∨, 0) and (Σ, ∧, 1) form commutative monoids, ∨ and ∧ distribute over each other, and combining two complementary elements with one binary operation yields the identity of the other.

Many physics students find that abstract algebra clarifies the symmetries underlying quantum systems.

During the seminar, she explained why the algebraic closure of the rational numbers is fundamental to understanding polynomial roots.

A Kleene algebra is a De Morgan algebra that also satisfies the inequation x∧∼x⩽y∨∼y for all x and y, where "∼" denotes the De Morgan involution.

In modern quantum group theory, the Hopf algebra provides the algebraic framework for describing symmetries that possess both multiplication and comultiplication structures.