1.4 Introduction to algebraic numbers and algebraic integers

Algebraic numbers and integers form the foundation of algebraic number theory. They're complex numbers that are roots of polynomials with rational coefficients. Algebraic integers are a special case, being roots of monic polynomials with integer coefficients.

These concepts expand on familiar number systems like rationals and integers. They're crucial for understanding more advanced topics in algebraic number theory, like number fields and their properties. Mastering these basics opens doors to deeper mathematical insights.

Algebraic Numbers and Integers

Definitions and Examples

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• Algebraic numbers constitute complex numbers serving as roots of non-zero polynomial equations with rational coefficients
  • Expressed as solutions to equations $a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0$, where all $a_i$ are rational and $a_n \neq 0$
• Algebraic integers represent algebraic numbers functioning as roots of monic polynomials with integer coefficients
  • Monic polynomial features a leading coefficient (nonzero coefficient of highest degree) equal to 1
• Algebraic numbers encompass rational numbers, roots of rational numbers ($\sqrt{2}$, $\sqrt[3]{3}$), and complex roots of polynomials with rational coefficients ($i$, $-1+i\sqrt{3}$)
• Algebraic integers include integers, roots of unity ($i$, $-1$, $e^{2\pi i/n}$), and algebraic numbers serving as roots of monic polynomials with integer coefficients ($\sqrt{2}$, $(1+\sqrt{5})/2$)
• Transcendental numbers ($\pi$, $e$) fall outside the realm of algebraic numbers
• Set notation denotes algebraic numbers as $\mathbb{Q}$ (Q-bar) and algebraic integers as $\mathbb{O}$ or $A$

Set Properties and Relationships

• Algebraic numbers ($\mathbb{Q}$) form a superset of rational numbers ($\mathbb{Q}$)
• Algebraic integers ($\mathbb{O}$) encompass the set of integers ($\mathbb{Z}$)
• Field of algebraic numbers ($\mathbb{Q}$) represents the algebraic closure of rational numbers ($\mathbb{Q}$)
  • Contains all roots of polynomials with rational coefficients
• Ring of algebraic integers ($\mathbb{O}$) exhibits integral closure
  • Any algebraic number satisfying a monic polynomial equation with coefficients in $\mathbb{O}$ belongs to $\mathbb{O}$
• Intersection of algebraic integers ($\mathbb{O}$) and rational numbers ($\mathbb{Q}$) yields the set of integers ($\mathbb{Z}$)
  • Implies any rational algebraic integer constitutes an ordinary integer
• Field of fractions of the algebraic integers ring forms the algebraic numbers field
  • Analogous to the relationship between integers and rational numbers

Properties of Algebraic Numbers and Integers

Closure Properties

• Algebraic numbers exhibit closure under addition, subtraction, multiplication, and division (except by zero)
  • Arithmetic operations on algebraic numbers invariably produce another algebraic number
• Algebraic integers demonstrate closure under addition, subtraction, and multiplication
  • Division of algebraic integers does not necessarily yield an algebraic integer
• Closure property proofs utilize the fact that algebraic numbers and integers serve as roots of polynomials
  • Construction of new polynomials for arithmetic operation results
• Minimal polynomial of an algebraic number plays a crucial role in property proofs
  • Unique monic irreducible polynomial of least degree with the number as a root
• Sum and product of algebraic integers yield algebraic integers
  • Provable using resultants of polynomials or constructing appropriate monic polynomials with integer coefficients
• Conjugates of an algebraic number (other roots of its minimal polynomial) also qualify as algebraic numbers
  • Frequently employed in proofs involving algebraic numbers

Advanced Concepts

• Norm and trace of algebraic numbers facilitate problem-solving in divisibility and factorization within algebraic number fields
• Integral basis concept aids in understanding the structure of algebraic integers in number fields
• Discriminant of an algebraic number field provides crucial information about its arithmetic properties
• Galois theory applications elucidate the relationship between algebraic numbers and field extensions

Algebraic Numbers, Integers, and Rationals

Relationships and Distinctions

• Rational numbers ($\mathbb{Q}$) form a subset of algebraic numbers ($\mathbb{Q}$)
  • Every rational number serves as a root of a linear polynomial with integer coefficients
• Integers ($\mathbb{Z}$) constitute a subset of algebraic integers ($\mathbb{O}$)
  • Every integer functions as a root of a monic linear polynomial with integer coefficients
• Algebraic numbers ($\mathbb{Q}$) represent the algebraic closure of rational numbers ($\mathbb{Q}$)
  • Contain all roots of polynomials with rational coefficients
• Algebraic integers ($\mathbb{O}$) exhibit integral closure
  • Any algebraic number satisfying a monic polynomial equation with coefficients in $\mathbb{O}$ belongs to $\mathbb{O}$
• Intersection of algebraic integers ($\mathbb{O}$) and rational numbers ($\mathbb{Q}$) yields integers ($\mathbb{Z}$)
  • Any rational algebraic integer qualifies as an ordinary integer
• Field of fractions of the algebraic integers ring forms the algebraic numbers field
  • Mirrors the relationship between integers and rational numbers

Theoretical Foundations

• Fundamental Theorem of Algebra proves that every non-constant polynomial with complex coefficients has at least one complex root
  • Implies that the complex numbers form an algebraically closed field
• Primitive element theorem states that every finite separable field extension is simple
  • Crucial for understanding the structure of algebraic number fields
• Kronecker-Weber theorem establishes that every abelian extension of the rational numbers constitutes a subfield of a cyclotomic field
  • Connects algebraic number theory with cyclotomic fields and class field theory

Applications of Algebraic Numbers and Integers

Problem-Solving Techniques

• Determine algebraic or transcendental nature of numbers by seeking polynomials with rational coefficients having the number as a root
• Construct minimal polynomials for algebraic numbers to study properties and perform calculations
• Employ algebraic integer properties to factor polynomials over integers and find integral solutions to Diophantine equations
• Apply algebraic integer concepts to study rings of integers in algebraic number fields
  • Crucial for understanding the arithmetic of these fields
• Utilize norm and trace of algebraic numbers to solve problems involving divisibility and factorization in algebraic number fields
• Employ algebraic number and integer theory techniques to prove classical number theory results
  • Fundamental Theorem of Algebra
  • Properties of cyclotomic fields
• Apply algebraic number and integer concepts to cryptography
  • Construction and analysis of public-key cryptosystems based on algebraic number fields

Advanced Applications

• Utilize algebraic number theory in coding theory for error-correcting codes
• Apply algebraic integers in the study of Diophantine approximation and transcendence theory
• Employ algebraic number fields in the construction of elliptic curves for cryptography and number theory
• Use algebraic number theory techniques in solving Hilbert's problems
  • Particularly relevant for problems related to algebraic number fields and class field theory