Abstract Algebra

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

Type

Book chapter

Intellectual base

Textbook

Year

2004

ISBN

9780471433347

Link

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

Cited from

mathematics

Cited by

21 citations across 21 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.

• 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.

• 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.

• 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..
• 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 \(f_1\mid f_2\mid\cdots\mid f_r\).
• Irreducible polynomial
  • The coefficient domain is part of the predicate: the same expression can be irreducible over one field and reducible over an extension.
• Module (Algebra)
  • Standard algebra references use modules as the common language for ideals, abelian groups, linear representations, and homological constructions.
• 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

• 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.

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Links previously used in the corpus

Before the registry existed this work was also linked 4 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/Abstract-algebra/oclc/300249784 ×1

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