9.2 Integral Domains and Fields

Integral domains and fields are crucial structures in ring theory. They build on the concept of rings, adding special properties that make them powerful tools in algebra. Integral domains have no zero divisors, while fields allow division by any non-zero element.

These structures connect to broader algebraic concepts and have real-world applications. Understanding their properties and relationships helps us grasp more complex mathematical ideas and solve problems in various fields, from cryptography to coding theory.

Integral Domains and Fields

Definitions and Key Properties

• Integral domain constitutes a commutative ring with unity lacking zero divisors
• Field encompasses a commutative ring with unity where every non-zero element possesses a multiplicative inverse
• Cancellation law applies in integral domains (if $ab = ac$ and $a \neq 0$, then $b = c$)
• Fields form a subset of integral domains
• Characteristic of an integral domain or field denotes the smallest positive integer $n$ where $n \cdot 1 = 0$, or 0 if no such integer exists
  • Characteristic plays a crucial role in determining the structure and properties of the domain or field
  • For fields of characteristic 0, the integers can be embedded as a subring
• In fields, each non-zero element generates the entire multiplicative group
  • This property ensures that every non-zero element has a multiplicative inverse
  • It also implies that fields have a rich algebraic structure

Advanced Concepts

• Prime fields serve as the smallest subfield of a given field
  • For characteristic 0, the prime field is isomorphic to the rational numbers
  • For characteristic $p$, the prime field is isomorphic to $\mathbb{Z}/p\mathbb{Z}$
• Algebraic elements in a field satisfy polynomial equations with coefficients in the field
  • An algebraic extension is formed by adjoining algebraic elements to a field
• Transcendental elements do not satisfy any polynomial equation over the field
  • Transcendental extensions are created by adjoining transcendental elements

Examples of Integral Domains and Fields

Common Number Systems

• Integer ring ($\mathbb{Z}$) exemplifies an integral domain but not a field
  • Lacks multiplicative inverses for non-unit elements
• Rational number field ($\mathbb{Q}$) represents both an integral domain and a field
  • Every non-zero rational number has a multiplicative inverse
• Real number field ($\mathbb{R}$) constitutes an integral domain and a field
  • Completeness property distinguishes it from rational numbers
• Complex number field ($\mathbb{C}$) forms an integral domain and a field
  • Algebraically closed, meaning every non-constant polynomial has a root

Specialized Structures

• Finite fields (Galois fields) exist for every prime power order
  • $GF(p^n)$ denotes a finite field with $p^n$ elements, where $p$ is prime and $n$ is a positive integer
• Polynomial ring over an integral domain inherits the integral domain property
  • $F[x]$, where $F$ is a field, is an integral domain but not a field
• Ring of $n \times n$ matrices over a field fails to be an integral domain for $n > 1$
  • Contains zero divisors (non-zero matrices whose product is the zero matrix)

Proving Integral Domains and Fields

Proof Techniques for Integral Domains

• Demonstrate commutativity, unity existence, and absence of zero divisors to prove integral domain status
  • Commutativity: Show $ab = ba$ for all elements $a$ and $b$
  • Unity: Identify an element $1$ such that $1a = a1 = a$ for all $a$
  • No zero divisors: Prove that if $ab = 0$, then either $a = 0$ or $b = 0$
• Utilize counterexamples to disprove integral domain classification
  • Find two non-zero elements whose product is zero to show the existence of zero divisors

Proof Strategies for Fields

• Establish integral domain properties and existence of multiplicative inverses for all non-zero elements
  • For each non-zero element $a$, find an element $b$ such that $ab = ba = 1$
• Employ the fact that finite integral domains always constitute fields in proofs involving finite rings
  • This property stems from the pigeonhole principle applied to multiplication tables

Advanced Proof Techniques

• Incorporate ring characteristic in proofs, especially for finite fields
  • Use the fact that in a field of characteristic $p$, the equation $x^p = x$ holds for all elements
• Apply theorems about subrings and quotient rings to deduce properties of integral domains and fields
  • Subring of an integral domain is an integral domain
  • Quotient ring of an integral domain by a prime ideal is an integral domain

Relationship of Rings, Domains, and Fields

Hierarchical Structure

• Fields form a proper subset of integral domains
• Integral domains constitute a proper subset of commutative rings with unity
• Field of fractions of an integral domain represents the smallest field containing the integral domain
  • Constructed by forming ratios of elements from the integral domain
• Principal ideal domains encompass integral domains where every ideal is principal
  • Generated by a single element

Advanced Relationships

• Euclidean domains form a subset of principal ideal domains
  • Possess a Euclidean function allowing for a division algorithm
• Unique factorization domains include all principal ideal domains
  • Every non-zero non-unit element factors uniquely into irreducibles
• Quotient field of an integral domain exhibits isomorphism to its field of fractions
  • Provides an alternative construction of the smallest field containing the domain
• Ring achieves field status if and only if its only ideals are $\{0\}$ and the entire ring
  • This characterization connects field properties to ideal theory
• Algebraic closure of a field constitutes the smallest algebraically closed field containing it
  • Every polynomial with coefficients in this field has a root in the field