overview

Ring → Ideal → Module → Field → Extension → Galois Theory

• 기초 대수 구조

• Group

  • Subgroup
  • Normal Subgroup
  • Quotient Group
  • Homomorphism / Isomorphism Theorems

  • Ring

  • Commutative Ring / Noncommutative Ring

  • Ring with Unity
  • Subring
  • Zero Divisors / Integral Domain
  • Field

• Ideal (아이디얼)

• 정의와 성질

  • Additive Subgroup
  • Absorption Law

  • 종류

  • Left / Right / Two-sided Ideal

  • Principal Ideal
  • Prime Ideal
  • Maximal Ideal
  • Radical / Nilradical / Jacobson Radical
  • Primary Ideal

  • 연산

  • Sum / Product / Intersection

  • Colon Ideal \((I : J)\)
  • Ideal Quotient

  • 구조적 특성

  • Ideal Generated by a Set

  • Principal Ideal Domain (PID)
  • Noetherian Ring

  • 동치와 몫

  • Congruence Modulo Ideal

  • Quotient Ring \(R/I\)
  • Homomorphism & First Isomorphism Theorem

  • 특수 맥락

  • Polynomial Ring에서의 Ideal

  • Multivariate Ideal / Gröbner Basis
  • Hilbert Basis Theorem
  • Affine Variety와 Ideal의 대응 (Nullstellensatz)

• Module (가군)

• \(R\)-Module 정의

• Submodule
• Quotient Module
• Free / Finitely Generated Module
• Exact Sequence
• Hom and Tensor Functor
• Noetherian Module

• Field Theory

• Field Extension

  • Simple Extension
  • Algebraic vs Transcendental
  • Algebraic Closure
  • Degree of Extension
  • Tower Law
  • Minimal Polynomial
  • Splitting Field
  • Separable vs Inseparable Extensions

• Galois Theory

• Galois Extension

  • 정의: Normal + Separable
  • Fixed Field / Automorphism Group

  • Fundamental Theorem of Galois Theory

  • 중간체 ↔ 부분군 대응

  • Galois Group

  • \(\text{Gal}(E/F)\)

  • Solvable Group과 방정식의 해법

  • Application

  • 정다각형 작도 가능성

  • 방정식의 근의 대수적 표현 가능성
  • 고차 다항식의 근불가능성 증명

  • Examples

  • \(\mathbb{Q}(\sqrt{2})\)

  • \(\mathbb{Q}(\zeta_n)\)
  • Cyclotomic Extension
  • Finite Fields와 Galois Group