Key Definitions in Abstract Algebra

Group Theory

Basic Concepts

Group: An ordered pair $(G,\cdot )$ where $G$ is a non-empty set and $\cdot$ is a binary operation satisfying:

• Closure: $a\cdot b\in G$ for all $a,b\in G$
• Associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
• Identity: There exists $e\in G$ such that $e\cdot a=a\cdot e=a$
• Inverses: For each $a\in G$, there exists $a^{-1}\in G$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$

Abelian Group: A group where the operation is commutative: $a\cdot b=b\cdot a$ for all $a,b\in G$

Order of a Group: The number of elements in a finite group, denoted $|G|$

Order of an Element: The smallest positive integer $n$ such that $g^n=e$, denoted $|g|$

Subgroups

Subgroup: A non-empty subset $H$ of a group $G$ that forms a group under the same operation, denoted $H\le G$

Normal Subgroup: A subgroup $N$ of $G$ such that $gN{g}^{-1}=N$ for all $g\in G$, denoted $N⊴G$

Quotient Group: The group $G/N$ of cosets of a normal subgroup $N$ in $G$

Homomorphisms

Group Homomorphism: A map $\varphi :G\to H$ such that $\varphi (g_1{g}_2)=\varphi (g_1)\varphi (g_2)$

Isomorphism: A bijective homomorphism, denoted $G\cong H$

Kernel: $\ker (\varphi )=\{g\in G\mid \varphi (g)=e_H\}$

Image: $\text{Im}(\varphi )=\{\varphi (g)\mid g\in G\}$

Special Groups

Cyclic Group: A group generated by a single element, denoted $⟨g⟩$

Dihedral Group: The symmetry group of a regular $n$-gon, denoted $D_n$

Symmetric Group: The group of all permutations of $n$ elements, denoted $S_n$

Alternating Group: The subgroup of $S_n$ consisting of even permutations, denoted $A_n$

Simple Group: A group with no non-trivial normal subgroups

Ring Theory

Basic Concepts

Ring: A set $R$ with two operations $(+,\cdot )$ such that:

• $(R,+)$ is an abelian group
• $(R,\cdot )$ is a monoid
• Multiplication distributes over addition

Commutative Ring: A ring where multiplication is commutative

Ring with Identity: A ring containing a multiplicative identity element $1$

Zero Divisor: An element $a\ne 0$ such that $ab=0$ for some $b\ne 0$

Special Types of Rings

Integral Domain: A commutative ring with identity and no zero divisors

Field: A commutative ring with identity where every non-zero element has a multiplicative inverse

Division Ring: A ring with identity where every non-zero element has a multiplicative inverse (not necessarily commutative)

Ideals

Ideal: A subset $I$ of a ring $R$ such that:

• $(I,+)$ is a subgroup of $(R,+)$
• For all $r\in R$ and $x\in I$, both $rx$ and $xr$ are in $I$

Principal Ideal: An ideal generated by a single element, denoted $(a)$

Prime Ideal: A proper ideal $P$ such that if $ab\in P$, then either $a\in P$ or $b\in P$

Maximal Ideal: A proper ideal $M$ such that there is no ideal $I$ with $M\subset I\subset R$

Quotient Ring: The ring $R/I$ of cosets of an ideal $I$ in $R$

Special Domains

Principal Ideal Domain (PID): An integral domain where every ideal is principal

Euclidean Domain: An integral domain with a Euclidean function (division algorithm)

Unique Factorization Domain (UFD): An integral domain where every element has a unique factorization into irreducibles

Field Theory

Field Extensions

Field Extension: A pair of fields $L/K$ where $K$ is a subfield of $L$

Degree of Extension: The dimension of $L$ as a $K$-vector space, denoted $[L:K]$

Finite Extension: An extension of finite degree

Algebraic Element: An element $\alpha \in L$ that is a root of some non-zero polynomial in $K[x]$

Transcendental Element: An element that is not algebraic

Minimal Polynomial: The monic irreducible polynomial in $K[x]$ of which $\alpha$ is a root

Special Extensions

Normal Extension: An extension where every irreducible polynomial in $K[x]$ with a root in $L$ splits completely in $L$

Separable Extension: An extension where every element is separable (its minimal polynomial has no repeated roots)

Galois Extension: An extension that is both normal and separable

Splitting Field: The smallest field extension of $K$ in which a polynomial splits completely

Galois Theory

Galois Groups

Galois Group: The group of all automorphisms of a Galois extension $L/K$ that fix $K$, denoted $\text{Gal}(L/K)$

Fixed Field: The subfield of $L$ fixed by a subgroup $H$ of $\text{Gal}(L/K)$, denoted $L^H$

Solvability

Solvable Group: A group with a composition series whose factors are all abelian

Solvable by Radicals: A polynomial whose roots can be expressed using arithmetic operations and roots

Radical Extension: An extension obtained by adjoining roots

Module Theory

Modules

Module: An abelian group $M$ with a scalar multiplication $R\times M\to M$ satisfying certain axioms

Free Module: A module with a basis

Finitely Generated Module: A module generated by a finite set

Torsion Module: A module where every element has finite order

Torsion-Free Module: A module with no non-zero torsion elements

Polynomial Theory

Polynomials

Irreducible Polynomial: A polynomial that cannot be factored into non-constant polynomials

Primitive Polynomial: A polynomial whose coefficients have greatest common divisor 1

Content: The greatest common divisor of the coefficients of a polynomial

Eisenstein's Criterion: A sufficient condition for irreducibility of polynomials

Algebraic Geometry

Basic Concepts

Algebraic Set: The zero locus of a set of polynomials

Variety: An irreducible algebraic set

Coordinate Ring: The ring of polynomial functions on an algebraic set

Ideal of a Set: The ideal of polynomials vanishing on a set

Radical Ideal: An ideal equal to its own radical

Advanced Concepts

Infinite Galois Theory

Profinite Group: A topological group that is compact, Hausdorff, and totally disconnected

Krull Topology: The topology on an infinite Galois group

Transcendence Theory

Algebraically Independent: A set of elements that do not satisfy any non-trivial polynomial equation

Transcendence Basis: A maximal algebraically independent set

Transcendence Degree: The cardinality of a transcendence basis

Algebraic Closures

Algebraically Closed Field: A field where every non-constant polynomial has a root

Algebraic Closure: A minimal algebraically closed extension of a field

Historical Context

These definitions represent the evolution of abstract algebra from the 19th century to the present day. The concepts were developed to solve problems in:

• Number theory
• Geometry
• Analysis
• Cryptography
• Physics

The definitions provide a unified language for studying algebraic structures across mathematics and its applications.