1.4 Introduction to algebraic numbers and algebraic integers
5 min read•july 30, 2024
Algebraic numbers and integers form the foundation of 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 anxn+an−1xn−1+...+a1x+a0=0, where all ai are rational and an=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 (2, 33), and complex roots of polynomials with rational coefficients (i, −1+i3)
Algebraic integers include integers, (i, −1, e2πi/n), and algebraic numbers serving as roots of monic polynomials with integer coefficients (2, (1+5)/2)
Transcendental numbers (π, e) fall outside the realm of algebraic numbers
Set notation denotes algebraic numbers as Q (Q-bar) and algebraic integers as O or A
Set Properties and Relationships
Algebraic numbers (Q) form a superset of rational numbers (Q)
Algebraic integers (O) encompass the set of integers (Z)
Field of algebraic numbers (Q) represents the algebraic closure of rational numbers (Q)
Contains all roots of polynomials with rational coefficients
Ring of algebraic integers (O) exhibits
Any algebraic number satisfying a monic polynomial equation with coefficients in O belongs to O
Intersection of algebraic integers (O) and rational numbers (Q) yields the set of integers (Z)
Implies any rational 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
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
of an algebraic 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 (Q) form a subset of algebraic numbers (Q)
Every rational number serves as a root of a linear polynomial with integer coefficients
Integers (Z) constitute a subset of algebraic integers (O)
Every integer functions as a root of a monic linear polynomial with integer coefficients
Algebraic numbers (Q) represent the algebraic closure of rational numbers (Q)
Contain all roots of polynomials with rational coefficients
Algebraic integers (O) exhibit integral closure
Any algebraic number satisfying a monic polynomial equation with coefficients in O belongs to O
Intersection of algebraic integers (O) and rational numbers (Q) yields integers (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 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
Key Terms to Review (18)
Algebraic Integer: An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. These numbers include all integers and roots of unity, forming a crucial part of algebraic number theory, particularly when discussing the properties of number fields, norms, traces, minimal polynomials, and the structure of Dedekind domains.
Algebraic Number: An algebraic number is a complex number that is a root of a non-zero polynomial equation with rational coefficients. This concept plays a crucial role in number theory, particularly in understanding the structure of number fields and algebraic integers. Algebraic numbers include all rational numbers and roots of polynomials, which extends the notion of integers to a broader context in mathematics.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational contributions to various fields, including algebra, number theory, and mathematical logic. His work laid the groundwork for modern mathematics and significantly influenced the development of algebraic number theory.
Dedekind's Criterion: Dedekind's Criterion provides a way to determine whether a given number field is a Dedekind domain by examining the factorization of ideals in its ring of integers. This criterion connects algebraic integers, number fields, and the behavior of prime ideals within those fields, highlighting the relationship between algebraic structures and their integral bases.
Discriminant: The discriminant is a mathematical quantity that provides crucial information about the properties of algebraic equations, particularly polynomials. It helps determine whether a polynomial has distinct roots, repeated roots, or complex roots, which is essential for understanding the structure of number fields and their extensions.
Emmy Noether: Emmy Noether was a groundbreaking mathematician whose work significantly shaped modern algebra and number theory. She is best known for Noether's Theorem, which establishes a deep connection between symmetry and conservation laws in physics, but her contributions extend into the realm of algebraic integers and number fields, where she introduced innovative ideas about integral bases and ideal arithmetic. Her insights laid the foundation for many concepts that are essential for understanding advanced topics in algebra and number theory.
Field Extension: A field extension is a bigger field that contains a smaller field and allows for more solutions to polynomial equations. This concept helps in understanding how different fields relate to each other, especially when it comes to algebraic numbers, algebraic integers, and the properties of polynomials in those fields.
Finite Extension: A finite extension is a field extension in which the larger field has a finite dimension as a vector space over the smaller field. This concept connects various aspects of algebraic structures, showcasing how algebraic numbers and integers can form fields with finite degrees, and how properties such as norms, traces, and discriminants are integral to understanding these extensions.
Ideal Factorization: Ideal factorization refers to the process of expressing an ideal in a ring as a product of prime ideals, similar to how integers can be expressed as a product of prime numbers. This concept is crucial for understanding the structure of rings of integers and algebraic integers, where it reveals how ideals behave in relation to one another and how they can be decomposed within larger number fields or rings.
Infinite Extension: An infinite extension is a field extension that is not finitely generated, meaning it cannot be created from a finite number of elements. In the context of algebraic numbers and algebraic integers, infinite extensions play a crucial role in understanding the structure of number fields and their relationships with algebraic properties. These extensions allow for the exploration of more complex algebraic numbers that cannot be captured by finite means, thus deepening the understanding of the arithmetic of these number systems.
Integral Closure: Integral closure refers to the set of all elements in a given field that are integral over a specified ring, particularly focusing on algebraic integers. It connects various concepts like algebraic numbers and integers, providing a way to understand the structure of rings of integers in number fields, ensuring that algebraic properties are preserved within extensions.
Maximal Ideal: A maximal ideal is an ideal in a ring that is proper and maximal with respect to inclusion, meaning it cannot be contained in any larger proper ideal. This concept connects to important structures in algebraic systems, helping in the classification of rings and understanding algebraic integers and numbers, as well as paving the way to comprehend prime ideals and their unique properties within a ring.
Norm: In algebraic number theory, the norm of an algebraic number is a value that gives important information about its behavior in relation to a field extension. It can be viewed as a multiplicative measure that reflects how the number scales when considered within its minimal field, connecting properties of elements with their corresponding fields and extensions.
Number Field: A number field is a finite degree extension of the field of rational numbers, which means it is a larger set of numbers that includes rational numbers and is generated by adjoining algebraic numbers to the rationals. Number fields provide a framework for studying the properties of algebraic integers and their factorization, which connects deeply with various concepts in algebraic number theory.
Principal Ideal: A principal ideal is an ideal in a ring that can be generated by a single element. This concept is fundamental in algebraic structures, as it lays the groundwork for understanding more complex ideals and their relationships, particularly when working with algebraic numbers and integers. Principal ideals provide a way to connect ring theory with number theory, which is crucial when considering properties like uniqueness of factorization and the classification of ideals in rings of algebraic integers.
Quadratic Integers: Quadratic integers are a special class of algebraic integers that can be expressed in the form $$a + b\sqrt{d}$$, where $$a$$ and $$b$$ are integers, and $$d$$ is a square-free integer. These numbers extend the concept of integers to include solutions to polynomial equations with integer coefficients, specifically those of degree two. They play a critical role in number theory and can help form unique factorization domains.
Ring of Integers: The ring of integers is the set of algebraic integers in a number field, which forms a ring under the usual operations of addition and multiplication. This concept is crucial as it provides a framework for studying the properties and behaviors of numbers in various algebraic contexts, particularly when dealing with number fields, discriminants, and integral bases.
Roots of Unity: Roots of unity are complex numbers that satisfy the equation $$x^n = 1$$ for some positive integer $$n$$. They are significant in various areas of mathematics, particularly in understanding algebraic numbers and algebraic integers, as they exhibit properties that lead to the formulation of minimal polynomials, enabling a deeper exploration of field extensions and symmetry in algebraic structures.