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

• Binary Operation $\ast$
• Isomorphism
• Various Mappings
• Zero Morphism $0$

Group Theory

• Semigroup
• Monoid
• Group $G$
  • Left and Right Cancellation Laws
  • Uniqueness of Identity and Inverse Elements
  • Sock-Shoes Property $(ab)^{-1}=b^{-1}a^{-1}$
  • Commutator $[a,b]=aba^{-1}b^{-1}$
• Abelian Group

Finite Groups and Subgroups

• 🔒(25/11/10)Order $|G|$, $|g|$
• Subgroups $H \le G$
  • Subgroup Test
• 🔒(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 $\left< a \right>$
  • All Cyclic Groups are Abelian
  • Subgroups of a Cyclic Group are Cyclic
  • All Cyclic Groups are Isomorphic to the Integer Group
• Klein Four-Group $V$

Symmetric Groups

• Symmetric Group $S_A$
  • Dihedral Group $D_n$
  • No Permutation is Both Even and Odd
• Cayley’s Theorem
• Orbits, Cycles, and Transpositions
• Alternating Group $A_n$

Quotient Groups

• Cosets and Normal Subgroups $H \triangleleft G$
  • Properties of Cosets
• Lagrange’s Theorem
• Cartesian Product of Groups
• Kernel $\ker$
• Quotient Group $G/H$

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

• First Isomorphism Theorem
• Second Isomorphism Theorem
• Third Isomorphism Theorem

Ring Theory

• Ring $R$
  • Multiplicative Rules in Rings
  • Commutator $[a,b] = ab - ba$
  • Jacobi Identity $a \ast (b \ast c) + c \ast (a \ast b) + b \ast (c \ast a) = 0$
• Boolean Ring
  • If Zero Divisor in a Ring is Idempotent, It Can Be Expressed as a Direct Sum
• Noetherian Ring
• Dual Numbers
  • Differentiable Real Functions Defined on a Dual Numbers

Polynomials

• Ring of Polynomials $F[x]$
• Zeroes of Polynomial Functions
• Division Theorem
• Factor Theorem
• Irreducible Elements of Polynomial Functions
  • Eisenstein’s Criterion

Ideals

• Definition and Criterion for Subrings
• Ideal $I$
  • Ideal with a Unit Element
• 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

• R-Module
• F-Vector Space
• Graded Module

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 $E = F(\alpha)$
  • Algebraic Method of Constructing Complex Numbers from Real Numbers $\mathbb{R}[x] / \left< x^2 + 1 \right> \simeq \mathbb{C}$
• 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

• Algebraic and Transcendental Numbers
• Constructible Numbers
• 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