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
Cyclic Groups
Symmetric Groups
Quotient Groups
- Definition and Criterion for Subgroups
- 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
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
- 이원수
- 이원수 환 위에서 정의되는 미분가능한 실함수
- 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