Fields are fundamental algebraic structures that generalize familiar number systems. They provide a framework for studying polynomial equations and abstract algebra. Fields have deep connections to various branches of mathematics, including noncommutative geometry.
Fields are commutative rings with unity, satisfying properties like commutativity and associativity. They have unique multiplicative inverses for non-zero elements, allowing division. This makes fields suitable for solving equations and studying ratios in mathematics.
Definition of fields
Fields are fundamental algebraic structures that generalize the familiar number systems (rational, real, and complex numbers) and provide a framework for studying polynomial equations and abstract algebra
Fields have a rich theory and deep connections to various branches of mathematics, including noncommutative geometry, which studies geometric spaces and structures that do not necessarily satisfy the commutative property of multiplication
Commutative rings with unity
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Fields are commutative rings with unity, meaning they satisfy the following properties:
Commutativity of addition and multiplication: a+b=b+a and ab=ba for all elements a,b in the field
Associativity of addition and multiplication: (a+b)+c=a+(b+c) and (ab)c=a(bc) for all elements a,b,c in the field
Existence of additive and multiplicative identity elements: There exist elements 0 and 1 such that a+0=a and a⋅1=a for all elements a in the field
Existence of additive inverses: For every element a in the field, there exists an element −a such that a+(−a)=0
Distributivity of multiplication over addition: a(b+c)=ab+ac for all elements a,b,c in the field
Multiplicative inverses
In addition to the properties of commutative rings with unity, fields have the unique property that every non-zero element has a multiplicative inverse
For every non-zero element a in a field, there exists an element a−1 such that a⋅a−1=1
The existence of multiplicative inverses allows division to be well-defined in fields, making them suitable for solving equations and studying ratios
Axioms of fields
The axioms of fields can be summarized as follows:
(F,+) is an abelian group (satisfies commutativity, associativity, existence of identity, and existence of inverses for addition)
(F∖{0},⋅) is an abelian group (satisfies commutativity, associativity, existence of identity, and existence of inverses for multiplication)
Multiplication is distributive over addition: a(b+c)=ab+ac for all a,b,c∈F
These axioms provide a concise and abstract characterization of fields, allowing mathematicians to study their properties and relationships to other algebraic structures
Examples of fields
Several familiar number systems and abstract algebraic structures serve as examples of fields, each with its own unique properties and applications
Rational numbers
The set of rational numbers Q forms a field under the usual operations of addition and multiplication
Rational numbers are numbers that can be expressed as the ratio of two integers, i.e., a/b where a,b∈Z and b=0
The field of rational numbers is the smallest field containing the integers and is crucial in the study of number theory and algebra
Real numbers
The set of real numbers R is a field that extends the rational numbers and includes all points on the real line
Real numbers can be represented as infinite decimal expansions and are essential in calculus, analysis, and applied mathematics
The field of real numbers is complete, meaning that every Cauchy sequence of real numbers converges to a real number
Complex numbers
The set of complex numbers C is a field that extends the real numbers by introducing the imaginary unit i, satisfying i2=−1
Complex numbers are of the form a+bi, where a,b∈R, and are crucial in the study of complex analysis, algebraic geometry, and
The field of complex numbers is algebraically closed, meaning that every polynomial equation with complex coefficients has a solution in C
Finite fields
Finite fields, also known as Galois fields, are fields with a finite number of elements
The number of elements in a finite field is always a prime power, i.e., pn where p is a prime number and n is a positive integer
Examples of finite fields include:
The field of integers modulo a prime p, denoted Fp or Z/pZ
The field of polynomials over Fp modulo an irreducible polynomial of degree n, denoted Fpn
Finite fields have applications in cryptography, coding theory, and combinatorics
Field extensions
Field extensions are a fundamental concept in algebra that describe how one field can be "extended" to a larger field by adjoining new elements
The study of field extensions is essential in understanding the structure and properties of fields, as well as their relationships to polynomial equations and geometric constructions
Algebraic vs transcendental extensions
An extension field L of a field K is called an algebraic extension if every element of L is algebraic over K, meaning it is a root of some non-zero polynomial with coefficients in K
Example: The field of complex numbers C is an algebraic extension of the real numbers R, as every complex number is a root of a polynomial of the form x2+bx+c with b,c∈R
An extension field L of a field K is called a transcendental extension if there exists an element in L that is not algebraic over K
Example: The field of rational functions C(x) is a transcendental extension of the complex numbers C, as the variable x is not a root of any non-zero polynomial with complex coefficients
Degree of field extensions
The degree of a L/K is the dimension of L as a vector space over K, denoted [L:K]
For finite extensions, the degree can be calculated as the product of the degrees of intermediate extensions: if K⊆M⊆L, then [L:K]=[L:M][M:K]
The degree of an extension provides information about its complexity and the number of conjugate elements in the extension
Splitting fields
A splitting field of a polynomial f(x) over a field K is the smallest field extension of K in which f(x) factors into linear factors
The splitting field of a polynomial is unique up to isomorphism and can be constructed by adjoining the roots of the polynomial to the base field
Splitting fields are important in the study of polynomial equations and the Galois theory of field extensions
Normal extensions
A field extension L/K is called a normal extension if every irreducible polynomial in K[x] that has a root in L splits completely in L[x]
Equivalently, L/K is normal if and only if L is the splitting field of some polynomial over K
Normal extensions are crucial in the development of Galois theory and the study of symmetries of field extensions
Separable extensions
A field extension L/K is called a separable extension if every element of L is separable over K, meaning it has distinct conjugate elements in an algebraic closure of K
Equivalently, L/K is separable if and only if every irreducible polynomial in K[x] that has a root in L has distinct roots in an algebraic closure of K
Separable extensions are essential in the study of Galois theory and the characterization of abelian extensions
Galois theory
Galois theory is a branch of algebra that studies the symmetries and relationships between field extensions and their automorphism groups
Developed by Évariste Galois in the early 19th century, Galois theory provides a powerful framework for understanding the solvability of polynomial equations and the structure of finite extensions
Automorphism groups of fields
An automorphism of a field K is a bijective homomorphism from K to itself, i.e., a function σ:K→K such that:
σ(a+b)=σ(a)+σ(b) for all a,b∈K
σ(ab)=σ(a)σ(b) for all a,b∈K
σ is bijective (one-to-one and onto)
The set of all automorphisms of a field K forms a group under composition, called the automorphism group of K, denoted Aut(K)
The automorphism group of a field extension L/K consists of all automorphisms of L that fix K pointwise, denoted Gal(L/K) and called the Galois group of the extension
Fixed fields
For a field extension L/K and a subgroup H of the Galois group Gal(L/K), the fixed field of H is the set of all elements in L that are fixed by every automorphism in H:
LH={a∈L:σ(a)=a for all σ∈H}
The fixed field LH is a subfield of L containing K, and the degree of the extension L/LH is equal to the order of the subgroup H
The study of fixed fields and their relationships to subgroups of the Galois group is a central theme in Galois theory
Fundamental theorem of Galois theory
The fundamental theorem of Galois theory establishes a one-to-one correspondence between the subfields of a Galois extension L/K and the subgroups of its Galois group Gal(L/K)
Specifically, for a Galois extension L/K, the theorem states:
The map H↦LH is a bijection between the subgroups of Gal(L/K) and the subfields of L containing K
For subgroups H1,H2 of Gal(L/K), H1⊆H2 if and only if LH2⊆LH1
For a subgroup H of Gal(L/K), the extension L/LH is Galois and its Galois group is isomorphic to H
For a subfield M of L containing K, the extension L/M is Galois if and only if M=LH for some subgroup H of Gal(L/K)
The fundamental theorem of Galois theory provides a powerful tool for understanding the structure of field extensions and their symmetries
Solvability by radicals
A polynomial equation is said to be solvable by radicals if its roots can be expressed using the coefficients of the polynomial, the four arithmetic operations (addition, subtraction, multiplication, and division), and the extraction of n-th roots (square roots, cube roots, etc.)
Galois theory provides a criterion for determining the solvability by radicals of a polynomial equation based on the properties of its Galois group:
A polynomial equation is solvable by radicals if and only if its Galois group is a solvable group, meaning it has a composition series with abelian factor groups
The solvability by radicals of polynomial equations has important implications in the history of algebra and the development of abstract algebra
Insolvability of quintic equations
One of the most famous results in the history of algebra is the insolvability of general quintic equations (polynomial equations of degree 5) by radicals
Using Galois theory, it can be shown that the Galois group of a general quintic equation is the symmetric group S5, which is not a solvable group
Consequently, there exists no general formula involving only arithmetic operations and radicals that can express the roots of every quintic equation
The insolvability of quintic equations marked a turning point in the development of algebra and led to the emergence of abstract algebra and the study of algebraic structures beyond the realm of polynomial equations
Fields in noncommutative geometry
Noncommutative geometry is a branch of mathematics that generalizes the concepts of classical geometry to spaces and structures where the commutative property of multiplication may not hold
In this context, fields are replaced by more general algebraic structures called skew fields or division rings, which retain most of the properties of fields except for the commutativity of multiplication
Skew fields (division rings)
A skew field, also known as a division ring, is a ring D with unity in which every non-zero element has a multiplicative inverse
Skew fields satisfy all the axioms of fields except for the commutativity of multiplication, i.e., there may exist elements a,b∈D such that ab=ba
Examples of skew fields include:
The field of real numbers R and the field of complex numbers C (which are commutative)
The ring of quaternions H (which is non-commutative)
The study of skew fields is central to noncommutative geometry and the generalization of algebraic and geometric concepts to non-commutative settings
Quaternions
The quaternions, denoted H, form a 4-dimensional algebra over the real numbers, generated by the elements 1,i,j,k satisfying:
i2=j2=k2=−1
ij=k, jk=i, ki=j
ji=−k, kj=−i, ik=−j
Quaternions are a non-commutative skew field, extending the complex numbers and providing a framework for studying 3-dimensional rotations and special relativity
The algebra of quaternions has applications in computer graphics, control theory, and theoretical physics
Octonions
The octonions, denoted O, form an 8-dimensional algebra over the real numbers, extending the quaternions and generated by the elements 1,e1,e2,…,e7 satisfying specific multiplication rules
Octonions are a non-associative and non-commutative algebra, meaning that the associative property (ab)c=a(bc) and the commutative property ab=ba do not hold in general
The algebra of octonions is the largest normed division algebra and has connections to exceptional Lie groups and the classification of finite simple groups
Noncommutative field extensions
In noncommutative geometry, the concept of field extensions is generalized to the study of skew field extensions and noncommutative algebras
A extension is an algebra A over a skew field D such that A is a skew field containing D
The study of noncommutative field extensions involves the use of techniques from ring theory, module theory, and homological algebra
Noncommutative field extensions have applications in the study of , noncommutative algebraic geometry, and the Langlands program
Applications of fields
The theory of fields and their extensions has numerous applications across various branches of mathematics, as well as in science and engineering
Algebraic number theory
Algebraic number theory is the study of algebraic structures arising from the arithmetic of number fields, which are finite extensions of the field of rational numbers Q
The study of algebraic integers, prime ideals, and the factorization of ideals in number fields relies heavily on the properties of fields and their extensions
Important results in algebraic number theory, such as the Fundamental Theorem of Arithmetic for number fields and the Dirichlet Unit Theorem, are derived using the tools of field theory and Galois theory
Cryptography
Cryptography is the study of techniques for secure communication and data protection, and it relies on the properties of finite fields and algebraic structures
Many modern cryptographic protocols, such as the Advanced Encryption Standard (AES) and the Elliptic Curve Digital Signature Algorithm (ECDSA), use finite fields and their arithmetic to provide security and integrity
The hardness of certain computational problems in finite fields, such as the Discrete Logarithm Problem and the Ellip
Key Terms to Review (18)
Alain Connes: Alain Connes is a French mathematician known for his foundational work in noncommutative geometry, a field that extends classical geometry to accommodate the behavior of spaces where commutativity fails. His contributions have led to new understandings of various mathematical structures and their applications, bridging concepts from algebra, topology, and physics.
C*-algebra: A c*-algebra is a complex algebra of bounded linear operators on a Hilbert space that is closed under the operation of taking adjoints and is also closed in the norm topology. This structure allows the integration of algebraic, topological, and analytical properties, making it essential in both functional analysis and noncommutative geometry.
Classical limit: The classical limit refers to the transition from quantum mechanics to classical physics, where quantum systems exhibit behavior that can be described by classical laws. This concept is crucial in understanding how quantum phenomena relate to classical theories and serves as a bridge between these two realms of physics. It highlights how, under certain conditions, quantum systems can approximate classical behavior, particularly as the scale of the system increases or when action scales become large compared to Planck's constant.
Division Property: The division property states that if you have a non-zero element in a field, you can divide any element in that field by this non-zero element to get another element in the same field. This property is crucial because it establishes that every non-zero element has a multiplicative inverse, meaning you can effectively 'undo' multiplication by that element. It ensures that fields are structured in a way that supports both addition and multiplication, allowing for consistent arithmetic operations.
Field Extension: A field extension is a mathematical construct that involves a pair of fields, where one field is contained within another, allowing for the introduction of new elements and operations. This concept helps to explore how different fields can relate to each other, especially in terms of solving equations and understanding algebraic structures. Field extensions are crucial for understanding concepts like algebraic numbers and the construction of various algebraic structures.
Field Representations: Field representations refer to the ways in which algebraic structures, specifically fields, can be represented as matrices or linear transformations. This concept connects fields to linear algebra, showcasing how operations within a field can be understood in terms of matrix operations, thus linking abstract algebraic ideas with concrete numerical representations.
Finite dimensional algebras: Finite dimensional algebras are algebraic structures that consist of a vector space over a field, equipped with a bilinear product that satisfies the properties of associativity and distributivity. The dimension of the algebra, which is the size of its basis, is finite, meaning it contains a limited number of elements that can be used to express any element in the algebra as a linear combination. This concept plays a crucial role in understanding representation theory and the structure of various algebraic systems.
Gelfand-Naimark Theorem: The Gelfand-Naimark Theorem is a fundamental result in functional analysis that establishes a deep connection between commutative C*-algebras and compact Hausdorff spaces. It states that every commutative C*-algebra can be represented as continuous functions on some compact Hausdorff space, revealing how algebraic structures relate to geometric and topological concepts.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and computer scientist who made significant contributions to various fields including functional analysis, quantum mechanics, game theory, and operator algebras. His work laid the groundwork for many concepts in mathematics and theoretical physics, particularly in relation to the algebraic structures that underpin quantum theory and noncommutative geometry.
K-theory: K-theory is a branch of mathematics that studies vector bundles and their generalizations through the use of algebraic topology and homological algebra. It provides a framework for understanding the structure of these bundles, allowing for the classification of topological spaces and algebras, which has deep implications in various mathematical fields, including geometry and number theory.
Mackey's Theorem: Mackey's Theorem is a significant result in representation theory that describes how representations of groups can be understood in terms of their actions on certain spaces. This theorem highlights the relationship between the algebraic structures of groups and the geometric structures of spaces they act upon, revealing important connections between representations and the underlying fields associated with them.
Noncommutative field: A noncommutative field is a type of algebraic structure where the multiplication operation does not satisfy the commutative property, meaning that for elements 'a' and 'b', it is not always true that 'a * b = b * a'. In this framework, the exploration of fields expands into areas where traditional notions of algebra are altered, leading to rich structures in mathematical theory. Noncommutative fields can be essential in areas like quantum mechanics and noncommutative geometry, impacting how we understand symmetry and structure in mathematics.
Operator Algebra: Operator algebra is a mathematical framework that studies algebraic structures consisting of bounded operators on a Hilbert space, focusing on their relationships and properties. This concept connects various branches of mathematics and physics, allowing for the exploration of spaces that are not necessarily commutative, which leads to insights in both analysis and geometry.
Quantum Groups: Quantum groups are algebraic structures that generalize the concept of groups and are essential in the study of noncommutative geometry and mathematical physics. They play a pivotal role in the representation theory of noncommutative spaces and provide a framework for understanding symmetries in quantum mechanics, connecting seamlessly to various concepts in geometry and algebra.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at the smallest scales, typically at the level of atoms and subatomic particles. It introduces concepts like superposition and entanglement, which challenge classical intuitions about how particles behave and interact. This theory has profound implications across various fields, including mathematics, where its principles influence structures like algebras and fields, as well as statistical methods used in probability.
Spectral Triples: Spectral triples are mathematical structures used in noncommutative geometry that generalize the notion of a geometric space by combining algebraic and analytic data. They consist of an algebra, a Hilbert space, and a self-adjoint operator, which together capture the essence of both classical geometry and quantum mechanics, making them a powerful tool for studying various mathematical and physical concepts.
String Theory: String theory is a theoretical framework in physics that posits that the fundamental particles of the universe are not point-like objects, but rather one-dimensional strings that vibrate at different frequencies. This idea suggests that the various properties of particles, such as mass and charge, arise from the different vibrational modes of these strings.
Von Neumann algebra: A von Neumann algebra is a type of operator algebra that is defined as a *-subalgebra of bounded operators on a Hilbert space which is closed in the weak operator topology and contains the identity operator. This structure plays a crucial role in the study of quantum mechanics and noncommutative geometry, particularly when discussing representations, integration, and differential calculus in infinite-dimensional spaces.