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Topics in an Abstract Algebra Course

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General

Abstract Algebra	Abstract algebra is the set of advanced topics in algebra that deal with abstract algebraic structures rather than the usual number systems.
Algebraic Variety	The zero set of a collection of polynomials. An algebraic variety is one of the the fundamental objects in algebraic geometry.
Boolean Algebra	A Boolean algebra is an algebra where the multiplication and addition also satisfy the properties of the AND and OR operations from logic.
Category	A category is an abstract mathematical object that generalizes the ideas of maps and commutative diagrams.
Isomorphism	An isomorphism is a map between mathematical objects such as groups, rings, or fields that is one-to-one, onto, and preserves the properties of the object.
Lie Algebra	A Lie algebra is a nonassociative algebra corresponding to a Lie group.
Lie Group	A Lie group is a differentiable manifold that has the structure of a group and that satisfies the additional condition that the group operations of multiplication and inversion are continuous.

Group Theory

Abelian Group:	An Abelian group is a group for which the binary operation is commutative.
Cyclic Group:	A cyclic graph is an (always Abelian) abstract group generated by a single element.
Dihedral Group:	The dihedral group of order n>i is the symmetry group for a regular polygon with n sides.
Finite Group:	A finite group is a group with a finite number of elements.
Group:	A mathematical group is a set of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.
Group Action:	A group action is the association of each of a mathematical group's elements with a permutation of the elements of a set.
Group Representation:	A group representation is a mathematical group action on a vector space.
Group Theory:	Group theory is the mathematical study of abstract groups, namely sets of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.
Normal Subgroup:	A normal subgroup is a subgroup that is fixed under conjugation by any element.
Simple Group:	A simple group is a mathematical group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group.
Subgroup:	A subgroup is a subset of a mathematical group that is also a group.
Symmetric Group:	A symmetric group is a group of all permutations of a given set.
Symmetry Group:	A symmetry group is a group of symmetry-preserving operations, i.e., rotations, reflections, and inversions.

Rings and Fields

Algebra:	(1) Algebra is a subject taught in grade school and high school, sometimes referred to as "arithmetic", that includes the solution of polynomial equations in one or more variables and basic properties of functions and graphs. (2) In higher mathematics, the term algebra generally refers to abstract algebra, which involves advanced topics that deal with abstract algebraic structures rather than the usual number systems. (3) In topology, an algebra is a vector space that also possesses a vector multiplication.
Algebraic Number:	An algebraic number is a number that is the root of some polynomial with integer coefficients. Algebraic numbers can be real or complex and need not be rational.
Field:	A field is a ring in which every nonzero element has a multiplicative inverse. The real numbers and the complex numbers are both fields.
Finite Field:	A finite field is a field with a finite number of elements. In such a field, the number of elements is always a power of a prime.
Gaussian Integer:	A Gaussian integer is a complex number a + b i, where a and b are integers and i is the imaginary unit.
Ideal:	In mathematics, and ideal is a subset of a ring that is closed under addition and multiplication by any element of the ring.
Module:	A module is a generalization of a vector space in which the scalars form a ring rather than a field.
Quaternion:	A quaternion is a member of a four-dimensional noncommutative division algebra (i.e., a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative) over the real numbers.
Ring:	In mathematics, a ring is an Abelian group together with a rule for multiplying its elements.