Modern Algebra 1

GroupPermalink

Lagrange Theorem

If $H$ is a subgrop of a group $G$ , then $| H | ∣ | G |$ , in other words, $| G | = [G : H] | H |$ .

Cyclic Group
Symmetric Group
Coset
Lagrange Theorem 🔥🔥
Permutation Group
orbit; cycle; transposition
Alternating Group

Fundamental Theorem of Finitely Generated Abelian GroupPermalink

F.T of f.g. Abelian Group

Every f.g. abelian group $G$ is isomorphic to direct product of cyclic groups.

G ≅ Z_{(p_{1})^{r_{1}}} \times Z_{(p_{2})^{r_{2}}} \times \dots \times Z_{(p_{n})^{r_{n}}} \times Z \times Z \times \dots \times Z

Where $p_{i}$ are primes, not necessarily distinct, and $r_{i}$ are positive integers.

Decomposable & Indecomposable group
$p$ -group ¹

Factor Group & HomomorphismPermalink

condition for operation well-definedness on Factor group

H ⊴ G ⟺ g H g^{- 1} \subseteq H (\forall g \in G)

Factor Group
- from normal subgroups
- from homomorphism
- Auto-morphism
  - inner automorphism $σ_{g}$

Fundamental Homomorphism Theorem

Let $ϕ : G ⟶ G^{'}$ be a group homomorphism, THEN

$ϕ [G]$ is a group.
$G / \ker ϕ ≅ ϕ [G]$

Fundamental Homomorphism Theorem(FHT); 1st Isomorphism Thm 🔥🔥
- Canonical homomorphism $γ : G ⟶ G / H (H ⊴ G)$
index-2 group is normal
Factor Group - Application
- Converse of Lagrange Thm

Advanced Group TheoryPermalink

Three Isomorphism Theorems 🔥
Three Sylow Theorems 🔥
- $p$ -group
- normalizer of $H$ in $G$ ; $N_{G} (H)$
- Sylow $p$ -group
- Application 1 🔥
- Application 2 🔥
- Examples 🔥
  - Type-1
  - Type-2

Ring & FieldPermalink

Ring 1, 2
- commutative ring
- Ring homormophism & isomorphism
- Unity; multiplicative identity
- division ring
- Field & Skew Field
- Quaternion
- zero-divisor
- Bezout’s Identity
- Charasteric of Ring; $Char (R)$
Integral Domain
- Field $⟹$ Integral Domain
- Finite Integral Domain $⟹$ Field

Fermat’s Little Theorem

Let $p$ be a prime, IF $a \in Z$ , and $p ∤ a$ , THEN

a^{p - 1} \equiv 1 (mod p)

Fermat’s Little Theorem 🔥
- Examples

Euler’s Theorem

If $a \in Z$ , and $(a, n) = 1$ , THEN

a^{φ (n)} \equiv 1 (mod p)

NOTE: if $n = p$ , then $φ (p) = p - 1$

Euler’s Theorem 🔥
- Euler phi function
- Application: solve modulo equation $a x \equiv b$ (mod $n$ )

Quotient Field (= Field of Qutients; 분수체)
- Extend integral domain $D$ into field $F$ .
  - Extend $Z$ into $Q$

Ring of Polynomials
- Evaluation Homomorphism; $ϕ_{α}$
- Division Algorithm for Polynomial Ring
- zero of polynomial
- Factor Theorem; 인수 정리
- cyclic group embedding

Eisenstein Criteria

Let $p \in Z$ be a prime, $f (x) = a_{n} x^{n} + \dots a_{0} \in Z [x]$

\begin{aligned} a_{n} & ≢ 0 (mod p) \\ a_{i} & \equiv 0 (mod p) 0 \leq i < n \\ a_{0} & ≢ 0 (mod p^{2}) \end{aligned}

THEN, $f (x)$ is irreducible over $Q$ .

Irreducible Polynomials
- Eisenstein Criteria 🔥
Non-commutative Examples
Group Rings & Group Algebras
- Group Ring; $R (G)$
- Group Algebra; $F (G)$
- The Quaternions; $H$
- Wedderburn’s Theorem
  - “Every finite division ring is a field.”

Factor Ring & IdealPermalink

Ring Homomorphism & Factor Ring
- Factor Ring well-definedness
- Ideal
Maximal & Prime Ideals
- Ideal + unity = Ring 🔥
- Maximal Ideal
  - Maximal Ideal makes factor group as field.
- Prime Ideal
  - Prime Ideal makes factor group as integral domain.
- Maximal Ideal implies Prime Ideal
Prime Field
- $Char$ 와 sub-ring / sub-field 사이의 관계
- $Z_{p}$ , $Q$ are Prime Field

Prime ideals generalize the concept of primality to more general commutative rings.

Prime & Irreducible element
- Prime element
- Irreducible element
Principal Ideal
- Principal Integral Domain; PID

Advanced Ring & Field TheoryPermalink

Unique Factorization Domain; UFD 1, 2
- Associate / Associated element 🔥
- In UFD, Irreducible elt is also a Prime elt.
- Every PID is UFD 🔥
Fundamental Theorem of Arithmetic
Gauss’s Lemma
- (primitive) $\times$ (primitive) = (primitive)
Polynomial over UFD
Euclidean Domain
- Euclidean norm
- Euclidean Algorithm
Gaussian Integers 🔥
- Multiplicative norm
Fermat’s Theorem on Sums of Two Squares; 페르마의 두 제곱수 정리

Galois TheoryPermalink

🔥 Continued on Morden Algebra II … 🔥

Problem SolvingPermalink

AppendixPermalink

For a homormophism $ϕ$ ,

if $\ker ϕ = {e}$ , then $ϕ$ is 1-1.

Ring-Domain-Field

Field $⟹$ Integral Domain
Finite Integral Domain $⟹$ Field
Finite Division Ring $⟹$ Field (Wedderburn’s Theorem)

헷갈리는 조합 1

Quotient Field
Factor Ring

헷갈리는 조합 2

Factor Ring Ring / Ideal
Factor Theorem
Unique Factorization Domain(UFD)

Homomorphism 모음

canonical homoomprhism (= natural homomorphism)
- (Group) $ϕ : G ⟶ G / N$
- (Ring) $ϕ : R ⟶ R / I$
evaluation homomorphism

Maximal Ideal 모음

$p Z$ is Maximal Ideal.
Factor Ring from Maximal Ideal = Field
$F [x]$ is a Field and $p (x) \in F [x]$
- $⟨ p (x) ⟩$ is a Maximal Ideal $i f f$ $p (x)$ is irreducible in $F [x]$ .

Study MaterialsPermalink

『A First Course in Abstract Algebra』 Fraleigh, 7th ed.
『Abstract Algebra』 Dummit & Foote, 3rd ed.

Sylow Theorem 할 때도 잠깐 나온다! ↩