Comprehensive Abstract Algebra and Number Theory Study Guide

Number Theory and Arithmetic Principles

• Inclusion-Exclusion Principle (Αρχή Εγκλεισμού / Αποκλεισμού)

  • For a finite collection of sets $A_1, A_2, \dots, A_n$, the size of their union is calculated as:         $\left| \bigcup_{i=1}^n A_i \right| = \sum_{1 \le i \le n} |A_i| - \sum_{1 \le i < j \le n} |A_i \cap A_j| + \sum_{1 \le i < j < k \le n} |A_i \cap A_j \cap A_k| - \dots + (-1)^{n-1} |A_1 \cap A_2 \cap \dots \cap A_n|$

• Bézout's Identity and Coprimality

  • Two integers $a$ and $b$ are coprime, denoted as $\gcd(a, b) = 1$, if and only if there exist integers $m, n \in \mathbb{Z}$ such that:         $1 = m \cdot a + n \cdot b$

• Euler's Totient Theorem

  • If $a$ and $n$ are positive integers such that $\gcd(a, n) = 1$, then:         $a^{\phi(n)} \equiv 1 \pmod n$         where $\phi(n)$ represents Euler's totient function.

• Fermat's Little Theorem and Congruence Cancellation

  • Fermat's Little Theorem: If $p$ is a prime number and $a \in \mathbb{Z}$, then:         $a^p \equiv a \pmod p$
  • If $p$ is a prime number and $a$ is an integer such that $p \nmid a$, then:         $a^{p-1} \equiv 1 \pmod p$
  • Cancellation Law in Modular Arithmetic: If $a \cdot c \equiv b \cdot c \pmod m$ with $c \neq 0$, then:         $a \equiv b \pmod{\frac{m}{d}}$         where $d = \gcd(c, m)$.

Ring Theory and Algebraic Structures

• Ring Isomorphisms and Invariant Properties

  • Two rings $A$ and $B$ are called isomorphic, denoted as $A \cong B$ (or $A \sim B$), if there exists a ring isomorphism between them. Isomorphic rings share identical algebraic properties, including:
    • a) Element Count: Equal cardinality / number of elements ($|A| = |B|$).
    • b) Algebraic Laws: Commutativity and associativity of operations.
    • c) Identity and Generators: Existence of a multiplicative identity element ($1_A$ and $1_B$) and structure of generating sets.
    • d) Field and Unit Properties: Field structure and equal number of invertible elements (units).
    • e) Zero Divisors and Integral Domains: Presence or absence of zero divisors, preserving integral domain status.
    • f) Characteristic: Equal ring characteristic, i.e., $\text{Char}(A) = \text{Char}(B)$.
    • g) Principal Ideal Property: Preservation of whether all ideals are principal.

• Subrings, Ring Centers, and Units

  • Subring Criteria: A subset $S$ of a ring $A$ ($S \subseteq A$) is a subring if and only if:
    1. $\forall x, y \in S \implies x - y \in S$
    2. $\forall x, y \in S \implies x \cdot y \in S$
  • Center of a Ring: The center of a ring $A$, denoted $C(A)$, is the set of elements that commute with every element in $A$:         $C(A) = \{a \in A \mid a \cdot x = x \cdot a, \forall x \in A\}$
  • Invertible Elements (Units): If $a \in A$ is an invertible element (unit), it possesses a multiplicative inverse $a^{-1} \in A$ such that a \cdot a^{-1} = a^{-1} \cdot a = 1_A$.\n\n* **Fields, Integral Domains, and Modular Integers**\n * **Integral Domain (Ακέραια Περιοχή):** A commutative ring with identity 1 \neq 0$that has no zero divisors (i.e., if$a \cdot b = 0$, then$a = 0$or$b = 0).\n * **Field Structure of Modular Integers:** The ring \mathbb{Z}p$is a field if and only if$p is a prime number.\n * **Field to Integral Domain Relationship:**\n * Every field is an integral domain.\n * The converse is not generally true; however, every **finite** integral domain is a field.\n\n* **Polynomial Rings, Associates, and Characteristic**\n * **Ring Characteristic Identity:** If A$is a ring with identity$1 \neq 0$, then the characteristic of the polynomial ring$A[x]$equals the characteristic of$A:\n        \text{Char}(A) = \text{Char}(A[x])\n * **Associates in Polynomial Rings:** Let A$be an integral domain and$f(x), g(x) \in A[x]$. If$f(x) \mid g(x)$and$g(x) \mid f(x)$, then$f(x)$and$g(x)$are associates, meaning there exists an invertible element (unit)$a \in A^\times such that:\n        f(x) = a \cdot g(x)\n\n* **Euclidean Algorithm for Polynomial Greatest Common Divisors**\n * To find the greatest common divisor \gcd(f(x), g(x))$of two polynomials$f(x), g(x) \in F[x]:\n 1. Perform successive Euclidean divisions.\n 2. Identify the last non-zero remainder in the chain of division steps.\n 3. Convert this last non-zero remainder into a monic polynomial (dividing by its leading coefficient).\n\n* **Rational Root Theorem**\n * Let f(x) = a_n x^n + a{n-1} x^{n-1} + \dots + a_1 x + a_0 \in \mathbb{Z}[x]$be a polynomial with integer coefficients. If a rational number$\frac{r}{s}$(written in lowest terms such that$\gcd(r, s) = 1$) is a root of$f(x), then:\n * r \mid a_0$($r$divides the constant term$a_0)\n * s \mid a_n$($s$divides the leading coefficient$a_n)\n\n* **Eisenstein's Irreducibility Criterion**\n * Let f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \in \mathbb{Z}[x]$be a polynomial with integer coefficients. If there exists a prime number$p such that:\n 1. p \nmid a_n$($p does not divide the leading coefficient)\n 2. p \mid a_i$for all$i \in {0, 1, \dots, n-1}$($p divides all other coefficients)\n 3. p^2 \nmid a_0$($p^2 does not divide the constant term)\n * Then the polynomial f(x)$is irreducible over the rational field$\mathbb{Q}[x].\n\n* **Ring Homomorphisms and Properties**\n * **Definition:** A mapping \phi: A \rightarrow B$between two rings$A$and$B$is called a ring homomorphism if for all$x, y \in A:\n 1. \phi(x + y) = \phi(x) + \phi(y)\n 2. \phi(x \cdot y) = \phi(x) \cdot \phi(y)\n * **Fundamental Properties:** For any ring homomorphism \phi: A \rightarrow B:\n 1. \phi(0_A) = 0_B\n 2. \phi(-a) = -\phi(a)$for all$a \in A\n 3. \phi(n \cdot a) = n \cdot \phi(a)$for all$n \in \mathbb{Z}$and$a \in A\n 4. \phi(a^n) = (\phi(a))^n$for all$n \in \mathbb{N}$and$a \in A\n\n* **Principal Ideals and Principal Ideal Domains (PID / Π.Κ.Ι.)**\n * **Principal Ideal:** Let A$be a commutative ring and$a \in A$. The principal ideal generated by$a$, denoted$\langle a \rangle, is defined as:\n        \langle a \rangle = {a \cdot x \mid x \in A}\n * **Principal Ideal Domain (PID / Περιοχή Κύριων Ιδεωδών):** A ring A is called a Principal Ideal Domain if:\n 1. A is commutative.\n 2. A$has a multiplicative identity$1 \neq 0$.
    3. $A$ has no zero divisors (it is an integral domain).
    4. Every ideal $I$ of $A$ is principal (i.e., $I = \langle a \rangle$ for some $a \in A$).
  • Polynomial PID Property: If $F$ is a field, then the polynomial ring $F[x]$ is a Principal Ideal Domain (PID).

• Quotient Rings and the First Isomorphism Theorem for Rings

  • Quotient Ring (Δακτύλιος Πηλίκου): Let $A$ be a ring and $I$ an ideal of $A$. The quotient ring $A/I$ is defined as:         $A/I = \{a + I \mid a \in A\}$         where cosets satisfy $a + I = b + I \iff a - b \in I$, equipped with operations:
    1. Addition: $(a + I) + (b + I) = (a + b) + I$
    2. Multiplication: $(a + I) \cdot (b + I) = (a \cdot b) + I$
  • First Isomorphism Theorem for Rings: Let $\phi: A \rightarrow B$ be a ring homomorphism with kernel $\text{Ker}(\phi) = \{a \in A \mid \phi(a) = 0_B\}$ and image $\text{Im}(\phi) = \{\phi(a) \mid a \in A\}$. Then:         $A / \text{Ker}(\phi) \cong \text{Im}(\phi)$

Group Theory Foundations

• Group Definition and Axioms

  • A set $G$ equipped with a binary operation $*$ is called a Group, denoted $(G, *)$, if it satisfies three axioms:
    1. Associativity: $(x * y) * z = x * (y * z)$ for all x, y, z \in G$.\n 2. **Identity Element:** There exists an element e \in G$such that$x * e = e * x = x$for all$x \in G$.
    2. Inverse Element: For every $x \in G$, there exists an element $y \in G$ (denoted $x^{-1}$) such that x * y = y * x = e$.\n * **Abelian Group:** A group G is called Abelian (or commutative) if:\n        x * y = y * x \quad \forall x, y \in G\n\n* **Group Order and Subgroup Criteria**\n * **Group Order:** The order of a group G$, denoted$|G|$, is the cardinal number representing the total count of elements in$G$.
  • Subgroup Definition and Criteria: A subset $H \subseteq G$ is a subgroup of $G$ (denoted $H \le G$) if:
    1. $e \in H$ (non-empty set condition)
    2. $\forall x, y \in H \implies x \cdot y \in H$ (closure under operation)
    3. $\forall x \in H \implies x^{-1} \in H$ (closure under inverses)
  • Equivalent Single Condition: A non-empty subset $H \subseteq G$ is a subgroup if and only if:         $\forall x, y \in H \implies x \cdot y^{-1} \in H$

• Product of Subgroups

  • Let $H$ and $K$ be subgroups of a group $G$. The product set $H \cdot K = \{h \cdot k \mid h \in H, k \in K\}$ is a subgroup of $G$ ($H \cdot K \le G$) if and only if:         $H \cdot K = K \cdot H$

• Center of a Group

  • The center of a group $G$, denoted $Z(G)$, is the set of elements that commute with all elements of $G$:         $Z(G) = \{g \in G \mid g \cdot x = x \cdot g, \forall x \in G\}$
  • The identity element $e$ always belongs to $Z(G)$.
  • A group $G$ is Abelian if and only if its center equals the whole group:         $G \text{ is Abelian} \iff Z(G) = G$

• Cyclic Groups, Element Orders, and Subgroup Properties

  • Cyclic Group: A group $G$ is cyclic if it is generated by a single element $x \in G$, denoted $G = \langle x \rangle = \{x^n \mid n \in \mathbb{Z}\}$.
  • Fundamental Properties of Cyclic Groups:
    • Every cyclic group is Abelian.
    • Every subgroup of a cyclic group is cyclic.
  • Order of an Element: The order of an element $x \in G$, denoted $|x|$, is equal to the order of the cyclic subgroup generated by $x$:         $|x| = |\langle x \rangle|$
  • Order of Powers in Finite Cyclic Groups: Let $G = \langle x \rangle$ be a finite cyclic group of order $|G| = m$. For any integer $k$, the order of $x^k$ is given by:         $|x^k| = \frac{m}{\gcd(m, k)}$
  • Generator Equivalence: In a cyclic group $G = \langle x \rangle$ of order $m$, the subgroups generated by powers $x^a$ and $x^b$ are equal if and only if:         $\langle x^a \rangle = \langle x^b \rangle \iff \gcd(a, m) = \gcd(b, m)$
  • Unique Subgroup Theorem for Cyclic Groups: Let $G = \langle x \rangle$ be a cyclic group of order $|G| = m$. For every positive divisor $d$ of $m$ ($d \mid m$), there exists a unique cyclic subgroup $H \le G$ of order $|H| = d$, given by:         $H = \langle x^{m/d} \rangle$

• Lagrange's Theorem

  • If $G$ is a finite group and $H$ is a subgroup of $G$ ($H \le G$), then the order of $H$ divides the order of $G$:         $|H| \mid |G|$

Group Homomorphisms, Normal Subgroups, and Isomorphism Theorems

• Properties of Group Homomorphisms

  • Let $f: G \rightarrow G'$ be a group homomorphism. Then the following properties hold:
    1. $f(e_G) = e_{G'}$
    2. $f(x^{-1}) = (f(x))^{-1}$ for all $x \in G$
    3. $f(x^m) = (f(x))^m$ for all $m \in \mathbb{Z}$ and $x \in G$
    4. If $H \le G$, then $f(H) \le G'$
    5. If $K \le G'$, then $f^{-1}(K) \le G$

• Kernel, Image, and Monomorphisms

  • Kernel: The kernel of $f: G \rightarrow G'$ is defined as:         $\text{Ker}(f) = \{x \in G \mid f(x) = e_{G'}\}$
  • Image: The image of $f: G \rightarrow G'$ is defined as:         $\text{Im}(f) = \{y \in G' \mid \exists x \in G, f(x) = y\}$
  • Monomorphism Condition: A homomorphism $f$ is a monomorphism (injective / 1-1) if and only if its kernel is trivial:         $f \text{ is 1-1} \iff \text{Ker}(f) = \{e_G\}$

• Normal Subgroups and Equivalent Conditions

  • A subgroup $H$ of $G$ is called a Normal Subgroup (denoted $H \triangleleft G$) if left cosets equal right cosets:         $x \cdot H = H \cdot x \quad \forall x \in G$
  • Equivalent Characterizations: For a subgroup $H \le G$, the following statements are equivalent:
    1. $H \triangleleft G$
    2. $x \cdot H = H \cdot x$ for all $x \in G$
    3. $x \cdot H \cdot x^{-1} \subseteq H$ for all $x \in G$
    4. $x \cdot H \cdot x^{-1} = H$ for all $x \in G$

• Simple Groups

  • A group $G$ is called a Simple Group (Απλή ομάδα) if its only normal subgroups are the trivial subgroups, namely $\mathbb{\{e\}}$ and $G$ itself.

• Quotient Groups

  • Let $H$ be a normal subgroup of $G$ ($H \triangleleft G$). The set of cosets $G/H = \{x \cdot H \mid x \in G\}$ forms a group called the Quotient Group (Ομάδα Πηλίκο) under coset multiplication:         $(x \cdot H) \cdot (y \cdot H) = (x \cdot y) \cdot H$

• The Four Isomorphism Theorems for Groups

  • First Isomorphism Theorem: If $f: G \rightarrow G'$ is a group homomorphism, then:         $G / \text{Ker}(f) \cong \text{Im}(f)$
  • Second Isomorphism Theorem: Let $H \le G$ and $K \triangleleft G$. Then:         $(H \cdot K) / K \cong H / (H \cap K)$
  • Third Isomorphism Theorem: Let $K \triangleleft G$, $H \triangleleft G$, and $K \triangleleft H$. Then:         $(G / K) / (H / K) \cong G / H$
  • Fourth Isomorphism Theorem (Correspondence Theorem): Let $K \triangleleft G$. There is a one-to-one (1-1) correspondence between the subgroups of $G$ containing $K$ and the subgroups of the quotient group $G/K$.