Abstract Algebra
Algebra, as is widely known, was a practical skill necessary for solving real-world problems. In modern mathematics, it stands as a major branch of pure mathematics and is still applied in practical fields such as Cryptography and Topological Data Analysis. While a typical mathematician can study other subjects without knowing algebra, knowing algebra can make many problems and concepts straightforward. Whether or not algebra suits one’s taste, it’s a subject that seems to compile common sense in the mathematical world, so studying it, even just the facts, is highly recommended.
Binary Operation Structures
Group Theory
Finite Groups and Subgroups
- 🔒(25/11/10)Order $|G|$, $|g|$
- Subgroups $H \le G$
- 🔒(25/11/12)Center of a Group $Z(G)$
- 🔒(25/11/14)Centralizer $C(a)$
Cyclic Groups
- Additive Group of Integer Modulo n $\mathbb{Z}_{n}$
- Cyclic Group $\left< a \right>$
- Klein Four-Group $V$
Symmetric Groups
Quotient Groups
- Cosets and Normal Subgroups $H \triangleleft G$
- Lagrange’s Theorem
- Cartesian Product of Groups
- Kernel $\ker$
- Quotient Group $G/H$
Lie Groups
- Lie Groups
- General Linear Group $\mathrm{GL}(n)$
- Special Linear Group $\mathrm{SL}(n)$
- Orthogonal Group $\mathrm{O}(n)$
- Unitary Group $\mathrm{U}(n)$
- Special Unitary Group $\mathrm{SU}(n)$
Group Actions
- Group Action
- Equivariant Map of Group Action
- Stabilizer Subgroup $G_x$
- Burnside’s Formula
- $p$-Groups
- Cauchy’s Theorem
- Sylow Theorems
Isomorphism Theorems
Ring Theory
Polynomials
- Ring of Polynomials $F[x]$
- Zeroes of Polynomial Functions
- Division Theorem
- Factor Theorem
- Irreducible Elements of Polynomial Functions
Ideals
- Definition and Criterion for Subrings
- Ideal $I$
- Maximal Ideal $M$
- Prime Ideal $P$
- Principal Ideal $\left< a \right>$
- Radical and Nilradical $\text{rad}$, $\text{nil}$
Integral Domains
- Zero Divisors and Integral Domains ID
- Principal Ideal Domain PID
- Unique Factorization Domain UFD $\gcd$
- Euclidean Domain ED
- Norm on an Integral Domain
- Bézout Domain
Modules
Algebraic Differentiation
- Field of Fractions and Function Fields
- Differential Ring $\left( R, d \right)$
- Differential Field $\left( F, \partial \right)$
- Ordinary Differential Ring and Partial Differential Ring $\left( R, \Delta \right)$
Field Theory
- Field $F$
- Extension Field and Kronecker’s Theorem $F \le E$
- Simple Extension Field $E = F(\alpha)$
- Vector Spaces in Abstract Algebra
- Algebraic Extension Field
- Fundamental Theorem of Algebra Expressed in Abstract Algebra Terms
- Prime Fields $\mathbb{Z}_p$, $\mathbb{Q}$
- Conjugate Isomorphism Theorem
- Automorphisms of Fields $\text{Auto}$
Three Classical Problems of Construction
Galois Theory
- Minimal Splitting Field
- Separable Extension Field
- Galois Field $\text{GF}(p^n)$
- Galois Theory
- Deriving the Quadratic Formula by Rote
- Solution of Cubic Equations
- Relationship Between the Roots and Coefficients of Quadratic/Cubic/n-th Degree Equations
References
- Fraleigh. (2003). A first course in abstract algebra(7th Edition)
- Sze-Tsen Hu. (1968). Introduction to Homological Algebra
- Hatcer. (2002). Algebraic Topology
All posts
- Derivation of the Quadratic Formula Step by Step
- Binary Operations in Abstract Algebra
- Semigroup in Abstract Algebra
- Monoids in Abstract Algebra
- In group Theory in Abstract Algebra
- Proof of the Law of Cosines
- Uniqueness Proof of Identity Elements and Inverse Elements in groups
- Commutative groups in Abstract Algebra
- Cyclic groups in Abstract Algebra
- Prove that All Cyclic groups are Abelian
- Isomorphism in Abstract Algebra
- Prove that a Subgroup of a Cyclic group is Cyclic
- Proving That All Cyclic groups are Isomorphic to the Integer group
- Symmetry groups in Abstract Algebra
- Infinite Cyclic groups in Abstract Algebra
- In English: Various Mappings in Abstract Algebra
- Klein Four-group
- Cayley's Theorem Proof
- Orbits, Cycles, and Permutations in Abstract Algebra
- Proof that a Permutation Cannot Be Both Even and Odd
- Alternating groups in Abstract Algebra
- Cosets and Normal Subgroups in Abstract Algebra
- Socks-Shoes Property: The Inverse of ab is Equal to the Product of the Inverse of b and the Inverse of a
- Definition and Test Method of Subgroups
- Proof of Lagrange's Theorem
- The Cartesian Product of groups
- Nucleus, Kernel in Abstract Algebra
- Quotient groups in Abstract Algebra
- group Actions
- Isotropic Subgroups
- Burnside's Lemma Derivation
- Proof of the First Isomorphism Theorem
- Proof of the Second Isomorphism Theorem
- Proof of the Third Isomorphism Theorem
- In P-groups in abstract algebra
- Proof of Cauchy's Theorem in group Theory
- Shilov's theorem
- Properties and Proofs of Surplus Types
- Definition and Criterion of Subrings
- Rings in Abstract Algebra
- Rules for Multiplication in a Ring
- Field Theory in Abstract Algebra
- Boolean Ring
- Reflection and Refraction
- If the Unit of a Ring is Idempotent, It Can Be Expressed as a Direct Sum
- Polynomial Rings
- Zeros of a Polynomial Function
- Division Theorem Proof
- Proof of the Factor Theorem
- Irreducible Elements of Polynomial Functions
- Eisenstein's Criterion
- Ideals in Abstract Algebra
- Radicals and Nilradicals in Abstract Algebra
- Units of an Ideal
- Maximal Ideal
- Covariant Ideals
- Main Ideals
- Definition and Proof of Kronecker's Theorem for Extension Bodies
- Algebraic Numbers and Transcendental Numbers
- Simple Enlargement Body
- Algebraic Methods to Construct the Field of Complex Numbers from the Field of Real Numbers
- Vector Spaces in Abstract Algebra
- Algebraic Extension
- The Fundamental Theorem of Algebra Expressed in Terms of Abstract Algebra
- Constructible Numbers
- Proof of the Three Classical Problems of Antiquity
- Solid State Physics
- Brain Ventricular Enlargement
- Principal Ideal Domain
- Unique Factorization Domain
- Euclidean Domain
- Proof of the Conjugate Isomorphism Theorem
- Automorphisms of a Body
- Minimum Splitting Field
- Scalable Divisible Body
- Galois Field
- Galois Theory
- Integral Domain Norm
- Formula for the Roots of a Cubic Equation
- Relationships between the Roots and Coefficients of Quadratic/Tertiary/nth Degree Equations
- What is a Commutator in Group Theory?
- What is a commutator in field theory?
- Binomial operation's Jacobi Identity
- Abstract Algebra in R-modules
- Lie Groups
- F-vector space in Abstract Algebra
- Zero Morphism
- Bezout's Theorem
- General Linear Group
- Unitary Group
- Special Linear Group
- Orthogonal Group
- Special Unitary Group
- Definition of a Grading Module
- Fraction Rings and Fraction Fields
- Differential Rings in Abstract Algebra
- Differential Fields in Abstract Algebra
- Partial Differential Rings and Differential Rings
- Additive Group of Integer Modulo n
- Dual Numbers
- Differentiable Real Functions Defined on a Dual Numbers
- Subgroup Test
- Equivariant Map of Group Action
- Representation of Groups
- Topological Group