Abstract Algebra

Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra. Wiley.

Type

Book

Intellectual base

Textbook

Year

2004

Pages cited

7-9

ISBN

9780471433347

Link

https://www.wiley.com/en-us/Abstract+Algebra%2C+3rd+Edition-p-9780471433347

Cited from

mathematics

Cited by

29 citations across 29 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Bijectivity
  • Three structural facts travel with the pattern. Counting transfer: a bijection between two collections proves they have the same size, even when neither is finite, making bijection a primary tool of cardinality reasoning. Inverse existence and uniqueness: a bijection guarantees a unique inverse, so operations and reasoning can flow in either direction. Composition closure: composing two bijections gives a bijection, and this stability under composition is what makes bijections the building blocks of symmetry groups, permutations, and reversible processes.

    This sourceEstablishes that bijections of a set compose to bijections and form the symmetric group, the basis for permutation and symmetry-group reasoning.

• Closure
  • Each successive structure layers further axioms onto a closed-under-operations carrier, and the closure axiom is what makes "iterate the operation and stay in the structure" a theorem rather than a wish, as Dummit and Foote (2003) develop in their canonical undergraduate treatment of the algebraic-structures pyramid.

    This sourceCanonical undergraduate/graduate-prep textbook developing the algebraic-structures pyramid (group, ring, field, module, vector space) with closure axioms layered into each successive structure.

    Supported in partVerified against the work's full text

    The publisher's page confirms Dummit and Foote's textbook develops algebraic structures for undergraduates, but says nothing about closure axioms or iteration theorems.

    “The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the student's understanding.”

    Read by an automated reader; not a human review. How support was checked

• Equivalence Relation
  • An equivalence relation is not the same as equality, a distinction that Dummit and Foote (2003) emphasise as the foundational axis along which coarser identifications generalise the equality relation.

    This sourceIntroduces equivalence relations and partitions in its preliminaries (Sec. 0.1) and uses them throughout (cosets, congruences, quotient structures); equality is the finest such relation, coarser equivalences generalizing it. (Note: the prime's prior annotation described 'the algebraic-structures pyramid with closure axioms layered into each structure'

• Factorization
  • And it bottoms out at a library of irreducibles — primes in integers, simple groups in finite groups, indecomposable representations in linear algebra, atomic mechanisms in a causal factorization — and the catalog of irreducibles is itself a deep structural fact about the system.

    This sourceTreats factorization into irreducibles — primes, irreducible polynomials, simple groups (Jordan–Hölder) — and unique-factorization domains as structural facts about an algebraic system.

• Identity Element
  • An identity element is a distinguished member of a carrier that leaves every other member unchanged under one stated operation: e • x = x • e = x, for every x with which e composes.

    This sourceGives the standard definition of an identity element relative to a set and one stated binary operation.

• Span
  • Linear algebra (canonical): the span of vectors is the set of all their linear combinations; the closure is a subspace, its dimension the rank, the gap between generators and span what makes change-of-basis useful. Group theory: a group generated by a set is the closure of that set under the group operation, the generated subgroup being the key structural object (Cayley graphs, presentations).

    This sourceSection 2.4 ("Subgroups Generated by Subsets of a Group") defines the subgroup generated by a subset as the closure of that set under the group operation (and inverses) — the smallest subgroup containing it — supporting the generated-subgroup-as-span reading (Cayley graphs, presentations).

Domain-specific

• Automorphism Group
  • This is one named output, not a definition of automorphism group generally.

    This sourceTreats group automorphisms and the automorphism groups of cyclic groups.

• Closed Set
  • In algebra, closure under an operation is the minimal requirement for a subset to inherit algebraic structure: the integers are closed under addition and multiplication (a subring of the rationals) but not under division

    This sourceThe standard algebra text, in which the integers are given as a subring of the rationals (Z a subring of Q) under the subring definition.

    Supported in partVerified against a saved copy of the source

    “2z is a subring of z, as is nz for any integer n”

    Read by an automated reader; not a human review. How support was checked

• Conjugacy class
  • The conjugacy class of an element is its orbit under the group's action on itself by conjugation.
• Direct sum of groups
  • … between constitutive identity and a convenient observable transfers from The direct sum of countably many copies of Z consists of integer sequences with only finitely many nonzero entries. to An algebraist distinguishes direct sum from unrestricted direct product and states internal versus external formulation..
• Euclidean domain
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Field of fractions
  • Equivalent polynomial pairs p/q form k(x), the smallest field containing the polynomial ring.
• Frobenius normal form
  • The Frobenius normal form, also called the invariant-factor rational canonical form, is the block diagonal matrix \(C(f_1)\oplus\cdots\oplus C(f_r)\) similar to \(A\), where the monic nonconstant invariant factors satisfy \(f_1\mid f_2\mid\cdots\mid f_r\).
• Image (of a Function)
  • In algebra more broadly, the image of a group or ring homomorphism is a subgroup or subring of the codomain, and the first isomorphism theorem identifies it as the quotient of the domain by the kernel

    This sourceThe image of a group or ring homomorphism is a subgroup or subring of the codomain, and the First Isomorphism Theorem gives G/ker φ ≅ φ(G).

    SupportedVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Irreducible polynomial
  • The coefficient domain is part of the predicate: the same expression can be irreducible over one field and reducible over an extension.
• Kernel
  • The kernel is not merely a bookkeeping set; it carries the full algebraic structure of a subobject: for a linear map it is a subspace of \(V\), for a group homomorphism a normal subgroup of \(G\), for a ring homomorphism an ideal of the ring

    This sourceReports, for a group homomorphism, that the kernel is the preimage of the identity and a normal subgroup of the domain. States the First Isomorphism Theorem for a homomorphism of groups.

    Supported in partVerified against the work's full text

    “the ideal [structure] of fields is trivial. Our approach to studying an [algebraic] [structure] through its”

    Read by an automated reader; not a human review. How support was checked

• Module (Algebra)
  • Standard algebra references use modules as the common language for ideals, abelian groups, linear representations, and homological constructions.
• Prime ideal
  • A prime ideal P is a proper ideal whose complement is multiplicatively closed, or equivalently whose quotient R/P has no nonzero zero divisors.
• Product of Rings
  • For finitely many factors the underlying direct sum and product sets coincide, but categorical coproduct and product remain different notions in unital rings.
• Ring Homomorphism
  • They make a ring map a structure-preserving function rather than an arbitrary correspondence.
• Rupture field
  • … the base field K, polynomial P and irreducibility status, chosen root a, generated extension K(a), quotient realization, extension degree and distinction from containing all roots are explicit. to An applied instance preserves the same invariant under changed scale, notation, dataset, jurisdiction, or implementation..
• Separable polynomial
  • … and characteristic, polynomial and degree, algebraic closure, distinct-root count, formal derivative, greatest common divisor and treatment of reducible or zero-derivative cases are explicit. to An applied instance preserves the same invariant under changed scale, notation, dataset, jurisdiction, or implementation..
• Splitting field
  • A splitting field of \(f\) over \(K\) is a field extension \(L/K\) in which \(f\) factors into linear factors and which is generated over \(K\) by the roots of \(f\).
• Subquotient
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Symmetric group
  • The word symmetric here names the full permutation group; it is not restricted to geometric symmetries that preserve a shape.

Mechanisms

• Composition Table
  • For a small arrow set it is the most direct realization of the composition rule the archetype needs.

    This sourceFor a small finite set with a binary operation, a Cayley table directly records the result of composing every ordered pair of elements.

• Group Action Model
  • The orbit-stabilizer relationship

    This sourceStates that the size of an element’s orbit equals the index of its stabilizer, allowing the number of distinct reachable configurations to be calculated.

• Homomorphism Check
  • Tracking the map's kernel

    This sourceIdentifies the kernel as exactly the equivalence collapsed by a homomorphism, with the quotient mapping isomorphically to its image.

• Structure-Preserving Map Specification
  • The named anchor is exact: a homomorphism is precisely a map that respects the operations, and an isomorphism adds invertibility on top.

    This sourceDefines homomorphisms as structure-preserving maps and isomorphisms as bijective, hence invertible, homomorphisms.

Verification

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Does it back the claim? Read against the text for 4 of 29 citations: 1 supported, 3 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

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Links previously used in the corpus

Before the registry existed this work was also linked 5 other ways.

• https://www.wiley.com/en-us/Abstract+Algebra,+3rd+Edition-p-9780471433347 ×4
• https://archive.org/details/abstractalgebra0000dumm_a8h6 ×1
• https://books.google.com/books?id=znzJygAACAAJ ×1
• https://search.worldcat.org/title/300249784 ×1
• https://search.worldcat.org/title/Abstract-algebra/oclc/300249784 ×1

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