Seokyun Ha

Kubernetes Lover
CS && MATH

Group

Lagrange Theorem

If $H$ is a subgrop of a group $G$, then $\lvert H \rvert \mid \lvert G \rvert$, in other words, $\lvert G \rvert = [G:H] \lvert H \rvert$.

Fundamental Theorem of Finitely Generated Abelian Group

F.T of f.g. Abelian Group

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

\[G \cong \mathbb{Z}_{(p_1)^{r_1}} \times \mathbb{Z}_{(p_2)^{r_2}} \times \cdots \times \mathbb{Z}_{(p_n)^{r_n}} \times \mathbb{Z} \times \mathbb{Z} \times \cdots \times \mathbb{Z}\]

Where $p_i$ are primes, not necessarily distinct, and $r_i$ are positive integers.

  • Decomposable & Indecomposable group
  • $p$-group 1

Factor Group & Homomorphism

condition for operation well-definedness on Factor group

\[H \trianglelefteq G \iff gHg^{-1} \subseteq H \quad (\forall g \in G)\]
  • Factor Group
    • from normal subgroups
    • from homomorphism
    • Auto-morphism
      • inner automorphism $\sigma_g$

Fundamental Homomorphism Theorem

Let $\phi: G \longrightarrow G’$ be a group homomorphism, THEN

  1. $\phi[G]$ is a group.
  2. ${G}/{\ker \phi} \cong \phi[G]$

Advanced Group Theory

Ring & Field

Fermat’s Little Theorem

Let $p$ be a prime, IF $a \in \mathbb{Z}$, and $p \nmid a$, THEN

\[a^{p-1} \equiv 1 \quad (\textrm{mod} \; p)\]

Euler’s Theorem

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

\[a^{\varphi(n)} \equiv 1 \quad (\textrm{mod} \; p)\]

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

  • Quotient Field (= Field of Qutients; 분수체)
    • Extend integral domain $D$ into field $F$.
      • Extend $\mathbb{Z}$ into $\mathbb{Q}$


Eisenstein Criteria

Let $p \in \mathbb{Z}$ be a prime, $f(x) = a_n x^n + \cdots a_0 \in \mathbb{Z}[x]$

IF

\[\begin{aligned} a_n &\not\equiv 0 \quad (\textrm{mod} \; p) \\ a_i &\equiv 0 \quad (\textrm{mod} \; p) \quad 0 \le i < n\\ a_0 &\not\equiv 0 \quad (\textrm{mod} \; p^2) \\ \end{aligned}\]

THEN, $f(x)$ is irreducible over $\mathbb{Q}$.

Factor Ring & Ideal

  • 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
    • $\textrm{Char}$와 sub-ring / sub-field 사이의 관계
    • $\mathbb{Z}_p$, $\mathbb{Q}$ are Prime Field

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

Advanced Ring & Field Theory

Galois Theory

🔥 Continued on Morden Algebra II … 🔥


Problem Solving


Appendix

For a homormophism $\phi$,

if $\ker \phi = \{ e \}$, then $\phi$ is 1-1.

Ring-Domain-Field

  • Field $\implies$ Integral Domain
  • Finite Integral Domain $\implies$ Field
  • Finite Division Ring $\implies$ 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) $\phi: G \longrightarrow G / N$
    • (Ring) $\phi: R \longrightarrow R / I$
  • evaluation homomorphism

Maximal Ideal 모음

  • $p\mathbb{Z}$ is Maximal Ideal.
  • Factor Ring from Maximal Ideal = Field
  • $F[x]$ is a Field and $p(x) \in F[x]$
    • $\left< p(x) \right>$ is a Maximal Ideal $iff$ $p(x)$ is irreducible in $F[x]$.

Study Materials

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

  1. Sylow Theorem 할 때도 잠깐 나온다! ↩