Glossary

1.2 Basic algebraic structures: groups, rings, and fields

4 min read•july 30, 2024

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, , identity, and inverse properties
    • Integers under form a
    • Non-zero rational numbers under form a group
  • Rings have two binary operations (typically addition and multiplication) satisfying specific axioms
    • Integers under addition and multiplication form a commutative
  • 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 (integers modulo p)

Substructures and properties

  • Subgroups, subrings, and subfields maintain the structure of the parent set under the same operations
  • refers to the number of elements
    • Finite groups have (positive integers)
    • Infinite groups have (integers, real numbers)
  • Abelian groups have a commutative binary operation
    • Integers under addition form an
    • 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)(a * b) * c = a * (b * c) for all elements a, b, c
  • in Abelian groups and commutative rings expressed as a∗b=b∗aa * b = b * a for all elements a and b
  • in rings and fields states a∗(b+c)=(a∗b)+(a∗c)a * (b + c) = (a * b) + (a * c) for all elements a, b, c
  • e in a group satisfies a∗e=e∗a=aa * e = e * a = a for all elements a
  • Inverse elements in groups satisfy a∗a−1=a−1∗a=ea * a^{-1} = a^{-1} * a = e, where e is the identity element

Advanced properties and laws

  • in groups states if a∗b=a∗ca * b = a * c or b∗a=c∗ab * a = c * a, then b=cb = c
  • in rings states if a∗b=0a * b = 0, then either a=0a = 0 or b=0b = 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 f satisfies f(a∗b)=f(a)∗f(b)f(a * b) = f(a) * f(b) for all elements a and b
  • Isomorphisms are bijective homomorphisms
    • Indicate identical algebraic structure between two structures
  • of a homomorphism contains elements mapping to the identity element in the codomain
    • Always a normal in group theory
  • of a homomorphism contains elements in the codomain mapped to by at least one element in the domain
    • Always a subgroup or

Theorems and applications

  • states for a f:G→Hf: G → H, G/ker(f)≅im(f)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 generalizes role of Z in relation to Q
    • Fundamental object in algebraic number theory

Advanced topics

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

Key Terms to Review (42)

Abelian Group: An abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying four key properties: closure, associativity, identity, and invertibility, along with the additional property of commutativity. This means that the order in which you combine elements does not matter, making abelian groups particularly important in many areas of mathematics. They serve as foundational structures in the study of more complex algebraic systems such as rings and fields.
Addition: Addition is a fundamental binary operation that combines two elements to produce a third element, typically representing the total or sum of those elements. This operation is essential in various algebraic structures, where it serves as a primary method for manipulating elements within groups, rings, and fields, each with specific properties and rules governing addition. It also plays a key role in number systems such as the Gaussian and Eisenstein integers, where unique characteristics of addition help define their algebraic structure.
Associativity: Associativity is a property that describes how operations combine elements in a set, specifically stating that when three or more elements are combined, the way in which they are grouped does not affect the outcome. This property is fundamental in various algebraic structures and helps ensure consistency in operations, allowing for simplification and rearrangement of expressions without changing their results.
Automorphism: An automorphism is a special type of isomorphism from a mathematical structure to itself that preserves the operations and relations of that structure. It highlights the symmetries within the structure, showing how its elements can be rearranged without changing the overall properties. Automorphisms play a significant role in understanding the internal structure of groups, rings, and fields, while also being central to concepts like Galois theory and normal extensions.
Cancellation Law: The cancellation law states that if a product of two elements in an algebraic structure equals a third element, then one of those elements can be 'canceled' under certain conditions to find the other. This concept is significant because it connects to how elements interact within structures like groups, rings, and fields, emphasizing the importance of the unique identities and operations defined in these systems.
Class Field Theory: Class field theory is a branch of algebraic number theory that studies the relationships between abelian extensions of number fields and the ideal class groups of those fields. It provides a powerful framework for understanding how number fields can be related to one another through their Galois groups, while also linking local properties at primes with global properties of the number field.
Commutativity: Commutativity is a fundamental property in algebraic structures that states the order of operations does not affect the outcome. In other words, for two elements a and b, the equation a * b = b * a holds true for an operation *. This property is essential as it simplifies calculations and allows for more flexible manipulation of expressions within algebraic systems.
Cyclic Group: A cyclic group is a type of group that can be generated by a single element, meaning that every element in the group can be expressed as a power of this generator. This structure is foundational in group theory, linking to the broader concepts of algebraic structures such as groups, rings, and fields. Cyclic groups can be either finite or infinite, and they exhibit many useful properties that help in understanding more complex algebraic structures.
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.
Distributive Property: The distributive property is a fundamental principle in algebra that states that multiplying a number by a sum is the same as multiplying each addend individually and then adding the results. This property is essential for simplifying expressions and solving equations, as it allows for the distribution of multiplication over addition or subtraction. Understanding this concept is vital for working with algebraic structures where operations can be expressed in terms of addition and multiplication, particularly in rings and fields.
Évariste Galois: Évariste Galois was a French mathematician known for his groundbreaking work in abstract algebra, particularly in the field of group theory and the theory of equations. His research laid the foundation for what is now called Galois theory, connecting field theory with group theory and revealing deep insights into the solvability of polynomial equations. Galois' ideas allow mathematicians to understand how the roots of polynomials are related to symmetry, leading to a powerful framework for studying algebraic structures.
Field: A field is a set equipped with two operations, addition and multiplication, that satisfy certain properties including commutativity, associativity, distributivity, and the existence of additive and multiplicative identities and inverses. Fields are crucial in algebra as they provide a structure where division is always possible except by zero, connecting to other algebraic structures like groups and rings through their operations and properties.
Finite Field: A finite field, also known as a Galois field, is a set equipped with two operations, addition and multiplication, satisfying the properties of a field, but containing a finite number of elements. This concept is fundamental in various mathematical disciplines, including algebraic structures where fields play a critical role, as well as in number theory and applications in coding theory and cryptography.
Finite order: Finite order refers to the property of an element within a mathematical structure, such as a group, where the element generates a finite cyclic subgroup. This means that when the element is repeatedly combined with itself under the group's operation, it will eventually return to the identity element after a finite number of operations. This concept is crucial as it highlights specific behaviors and characteristics of elements in algebraic structures like groups, rings, and fields.
First Isomorphism Theorem: The first isomorphism theorem states that if there is a homomorphism from one algebraic structure to another, the quotient of the domain by the kernel of the homomorphism is isomorphic to the image of the homomorphism. This theorem is crucial in understanding how different algebraic structures can be related through their homomorphic images, making it a foundational concept in group theory, ring theory, and field theory.
Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one complex root. This theorem connects the field of complex numbers with polynomial functions, ensuring that the number of roots corresponds to the degree of the polynomial. It is fundamental because it guarantees that polynomials can be fully factored over the complex numbers, which is crucial in understanding their behavior and properties.
Galois Theory: Galois Theory is a branch of mathematics that connects field theory and group theory, providing a framework to understand the symmetries of the roots of polynomial equations. It explores how the structure of field extensions relates to the properties of groups, especially focusing on the relationships between subfields and subgroups. This theory serves as a fundamental tool for determining when a polynomial can be solved by radicals and plays a crucial role in understanding the solvability of polynomial equations.
Group: A group is a set equipped with a binary operation that satisfies four fundamental properties: closure, associativity, identity, and invertibility. These properties allow elements of the group to be combined in a way that maintains the structure of the set. Groups are foundational in understanding more complex algebraic structures like rings and fields, as they provide a framework for studying symmetry and operations within mathematics.
Group homomorphism: A group homomorphism is a structure-preserving map between two groups that respects the group operation. It means that if you take two elements from the first group, combine them using the group operation, and then apply the homomorphism, it will yield the same result as if you had first applied the homomorphism to each element and then combined them in the second group. This concept is crucial in understanding how different groups relate to each other and plays a significant role in connecting various algebraic structures.
Group order: Group order is defined as the number of elements in a group. This fundamental concept is critical in understanding the structure and classification of groups, as it provides insight into their properties and behaviors. The order can affect various aspects of group theory, such as subgroup structure and the application of certain theorems, allowing for a deeper comprehension of how groups interact with one another.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or fields, that respects the operations defined on those structures. It allows us to transfer the properties and relations of one structure to another, facilitating the study of their similarities and differences. Homomorphisms play a crucial role in understanding the relationships between various algebraic systems and in proving important theorems related to their existence and uniqueness.
Ideal Theory: Ideal theory refers to a branch of abstract algebra that studies ideals, which are special subsets of rings that capture the notion of 'divisibility' and 'factorization' in a structured way. This concept connects to the historical development of number theory, influences the understanding of unique factorization in integers, and aligns with the foundational structures of groups, rings, and fields within algebra. The study of ideals allows mathematicians to generalize the properties of integers and explore relationships between various algebraic systems.
Identity element: An identity element is a special type of element in a set with an operation that, when combined with any other element in the set, leaves that element unchanged. This concept is crucial in understanding algebraic structures like groups, rings, and fields, where the identity element plays a key role in defining operations and properties of those structures. Each type of algebraic structure has its own identity element corresponding to its specific operation, which helps establish rules for interaction within that structure.
Image: In mathematics, particularly in algebraic structures, the term 'image' refers to the set of output values that a function or mapping produces from a given set of input values. The concept of image is crucial in understanding how elements from one set are transformed and related to another, especially when discussing homomorphisms between groups, rings, or fields, as it helps in revealing the structure-preserving characteristics of these mappings.
Infinite Order: Infinite order refers to an element in a group that does not have a finite number of applications of the group operation that result in the identity element. In simpler terms, if you keep applying the group operation to this element, you'll never return to the starting point, the identity. This concept is crucial when studying the structure of groups, as it highlights elements that behave differently than those of finite order, influencing how we understand the group's overall characteristics.
Integer Ring: An integer ring is a specific type of ring in abstract algebra, consisting of the set of integers $$ extbf{Z}$$ along with the operations of addition and multiplication. This structure satisfies properties such as closure, associativity, commutativity for both operations, and the existence of additive identities and inverses. The integer ring is fundamental in understanding more complex algebraic systems, particularly as it serves as an essential example of a commutative ring with unity and is vital in the study of unique factorization and ideals.
Inverse Element: An inverse element is a component in algebraic structures such that when it combines with a given element, it produces the identity element of that structure. In groups, the inverse of an element 'a' is denoted as 'a^{-1}', satisfying the equation 'a * a^{-1} = e', where 'e' is the identity element. This property ensures that every element in a group has an inverse, which is crucial for defining operations within the group and contributes to the overall structure of algebraic systems like rings and fields.
Inversion: Inversion refers to the process of finding an element in a set that, when combined with a given element using a specific operation, yields the identity element of that set. This concept is crucial in understanding the structure and behavior of algebraic systems, especially in groups, where each element must have an inverse for the group operation to satisfy certain properties. The idea extends to rings and fields, where inverses relate to both additive and multiplicative structures.
Isomorphism: Isomorphism refers to a structural similarity between two algebraic structures that allows for a one-to-one correspondence between their elements while preserving the operations defined on those structures. This concept is essential in various areas of mathematics, as it highlights the inherent equivalence between different algebraic systems, showing how they can behave identically despite potentially differing appearances.
Kernel: In mathematics, particularly in abstract algebra, the kernel is a fundamental concept that refers to the set of elements that are mapped to zero by a given linear transformation or homomorphism. This concept plays a crucial role in understanding the structure of algebraic systems, as it helps identify the relationships between different elements and reveals properties like injectivity and surjectivity of functions.
Lagrange's Theorem: Lagrange's Theorem states that in a finite group, the order of any subgroup divides the order of the group. This fundamental result highlights the relationship between the size of a group and its subgroups, emphasizing that the structure of groups is intimately connected to their divisors. Understanding this theorem lays the groundwork for exploring more complex properties of groups and their elements.
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.
Multiplication: Multiplication is a fundamental arithmetic operation that combines groups of equal sizes to find a total quantity. In the context of algebraic structures, it serves as a binary operation within groups, rings, and fields, allowing for the exploration of various properties such as associativity, commutativity, and the existence of multiplicative inverses. This operation also plays a critical role in defining specialized integers like Gaussian and Eisenstein integers, impacting how we understand divisibility and algebraic properties in these unique number systems.
Prime Ideal: A prime ideal in a ring is a special type of ideal such that if the product of two elements is in the ideal, then at least one of those elements must also be in the ideal. This concept helps to identify the structure and properties of rings, especially in understanding the relationships between ideals and factors in rings, connecting closely with notions like unique factorization.
Rational Field: The rational field, denoted as $$ ext{Q}$$, is the field consisting of all fractions formed by integers, where the denominator is not zero. This field is a fundamental example in algebra as it provides a structure where addition, subtraction, multiplication, and division (except by zero) are well-defined. The rational field connects to other algebraic structures by serving as a base for understanding more complex fields and playing a crucial role in the study of number systems.
Ring: A ring is a set equipped with two binary operations, typically called addition and multiplication, that satisfy certain properties such as associativity, distributivity, and the existence of an additive identity. Rings generalize familiar number systems, allowing us to explore their structures and relationships in more abstract algebraic contexts. This makes rings essential in understanding both algebraic systems and the fundamental results regarding existence and uniqueness within these frameworks.
Ring Isomorphism: A ring isomorphism is a structure-preserving map between two rings that shows they are essentially the same in terms of their algebraic properties. This means that not only does the function maintain the addition and multiplication operations, but it also ensures that there is a one-to-one correspondence between elements of both rings, making them indistinguishable from an algebraic standpoint. Understanding ring isomorphisms helps in identifying when two different rings can be treated as equivalent for purposes of analysis.
Subfield: A subfield is a subset of a field that itself is also a field under the same operations of addition and multiplication. It retains the structure of a field, meaning it must contain the additive and multiplicative identities, be closed under addition and multiplication, and every non-zero element must have a multiplicative inverse. Understanding subfields is crucial as they relate to the properties and behaviors of larger fields, which are foundational concepts in algebraic structures.
Subgroup: A subgroup is a subset of a group that is itself a group under the same operation as the larger group. This means that a subgroup must satisfy the group properties: closure, associativity, identity, and inverses, all while being contained within the original group. Understanding subgroups helps in analyzing the structure of groups and in recognizing how groups can be broken down into smaller, manageable pieces.
Subring: A subring is a subset of a ring that itself forms a ring under the same operations of addition and multiplication defined in the larger ring. To qualify as a subring, this subset must include the zero element, be closed under the ring operations, and contain the additive inverses of its elements. This concept is crucial because it helps understand how smaller algebraic structures can exist within larger ones, maintaining the properties that define rings.
Unit Group: The unit group of a ring is the set of elements that have multiplicative inverses within that ring. Understanding unit groups is crucial for exploring the structure of algebraic objects, particularly in relation to the behavior of integers and their generalizations, which often manifest in the study of rings and fields.
Zero Product Property: The zero product property states that if the product of two elements is zero, then at least one of those elements must be zero. This concept is essential in algebra because it helps in solving equations by allowing us to set each factor to zero individually. It connects deeply to the structure of rings, where the notion of multiplication is defined, and provides a foundational tool for analyzing solutions in fields, as these structures maintain this property.
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