1.2 Basic algebraic structures: groups, rings, and fields

Algebraic structures like groups, rings, and fields form the backbone of algebraic number theory. These abstract systems, with their specific operations and properties, provide a framework for understanding more complex mathematical concepts.

In this chapter, we'll explore these structures and their properties. We'll see how they relate to number theory and lay the groundwork for deeper study of algebraic number fields and their arithmetic properties.

Groups, rings, and fields

Fundamental definitions and examples

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• Groups consist of a set and a binary operation satisfying closure, associativity, identity, and inverse properties
  • Integers under addition form a group
  • Non-zero rational numbers under multiplication form a group
• Rings have two binary operations (typically addition and multiplication) satisfying specific axioms
  • Integers under addition and multiplication form a commutative ring
• Fields are rings where all non-zero elements have multiplicative inverses
  • Rational numbers, real numbers, and complex numbers are examples of fields
• Finite fields (Galois fields) contain a finite number of elements
  • Z/pZ for prime p represents the simplest finite field (integers modulo p)

Substructures and properties

• Subgroups, subrings, and subfields maintain the structure of the parent set under the same operations
• Group order refers to the number of elements
  • Finite groups have finite order (positive integers)
  • Infinite groups have infinite order (integers, real numbers)
• Abelian groups have a commutative binary operation
  • Integers under addition form an Abelian group
  • Invertible matrices under multiplication generally do not form an Abelian group

Properties of algebraic structures

Fundamental properties

• Associative property in groups states $(a * b) * c = a * (b * c)$ for all elements a, b, c
• Commutativity in Abelian groups and commutative rings expressed as $a * b = b * a$ for all elements a and b
• Distributive property in rings and fields states $a * (b + c) = (a * b) + (a * c)$ for all elements a, b, c
• Identity element e in a group satisfies $a * e = e * a = a$ for all elements a
• Inverse elements in groups satisfy $a * a^{-1} = a^{-1} * a = e$, where e is the identity element

Advanced properties and laws

• Cancellation law in groups states if $a * b = a * c$ or $b * a = c * a$, then $b = c$
• Zero product property in rings states if $a * b = 0$, then either $a = 0$ or $b = 0$ (or both)
  • Holds in integral domains but not in all rings
• Uniqueness of identity and inverse elements can be proven using group axioms
  • Essential exercise in group theory

Homomorphisms and isomorphisms

Definitions and basic concepts

• Homomorphisms preserve structure between algebraic structures
  • For groups, a homomorphism f satisfies $f(a * b) = f(a) * f(b)$ for all elements a and b
• Isomorphisms are bijective homomorphisms
  • Indicate identical algebraic structure between two structures
• Kernel of a homomorphism contains elements mapping to the identity element in the codomain
  • Always a normal subgroup in group theory
• Image of a homomorphism contains elements in the codomain mapped to by at least one element in the domain
  • Always a subgroup or subring

Theorems and applications

• First Isomorphism Theorem states for a group homomorphism $f: G → H$, $G/ker(f) ≅ im(f)$
  • Relates quotient groups to homomorphisms
• Automorphisms are isomorphisms from a structure to itself
  • Set of all automorphisms forms a group under composition
• Classification of finite simple groups relies on analysis of isomorphisms and automorphisms
  • Fundamental in advanced group theory

Algebraic structures in number theory

Foundational concepts

• Hierarchy of structures fundamental to algebraic number theory
  • Fields are rings, and rings are groups under addition
• Ring of integers Z forms the foundation for more complex structures
  • Number fields built upon this foundation
• Number fields are finite extensions of rational numbers Q
  • Core objects of study in algebraic number theory
• Ring of integers of a number field generalizes role of Z in relation to Q
  • Fundamental object in algebraic number theory

Advanced topics

• Ideal theory in rings crucial for understanding factorization properties
  • Particularly important in ring of integers of number fields
• Galois theory studies field extensions and their automorphism groups
  • Central to understanding structure of number fields
• Class field theory uses group theory to describe abelian extensions of number fields
  • Major achievement in algebraic number theory
• Study of units and unit group of ring of integers essential for arithmetic of number fields
  • Provides insights into fundamental properties of number fields