Algebra

Lang, S. (2002). Algebra. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2002

DOI

10.1007/978-1-4613-0041-0

ISBN

9781461265511

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/978-1-4613-0041-0

Cited from

mathematics

Cited by

24 citations across 24 artifacts.

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

Primes

• Equivalence Relation
  • Mathematics is the originating domain, with Lang (2002) presenting equivalence relations as the unifying formal device behind cosets, similarity classes, and quotient constructions across abstract algebra.

    This sourceComprehensive graduate algebra; develops cosets, congruences, similarity classes, and quotient constructions across groups, rings, modules, and fields with equivalence relations as the unifying device.

• Monoid
  • There is no third law: neither commutativity nor inverses is required.

    This sourceDefines a monoid as a set with an associative law of composition and a two-sided unit element, requiring neither commutativity nor inverses, and gives the additive and multiplicative monoids of the natural numbers.

Domain-specific

• Algebraically closed field
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Anticommutative property
  • For example, the distinction between constitutive identity and a convenient observable transfers from The Lie bracket satisfies [x,y]=-[y,x], making it anticommutative. to An algebraist distinguishes anticommutativity from alternation in characteristic two, where minus signs collapse..
• Coefficient
  • … example, the distinction between constitutive identity and a convenient observable transfers from In 2x squared minus x plus 3, the coefficients relative to powers of x are 2, minus 1 and 3. to A calculation distinguishes coefficients from parameters and notes that a basis change generally changes coefficient values..
• Conjugacy class
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Degree of a field extension
  • … instance directly demonstrates that the base-field inclusion is fixed and degree is the dimension of the extension as a vector space over that base, distinguished from transcendence degree. to An applied instance preserves the same invariant under a changed notation, scale, implementation, or empirical setting..
• Diagonal subgroup
  • … directly demonstrates that the group G, finite power and diagonal homomorphism are fixed and the subset contains exactly the constant-coordinate tuples with componentwise group operations. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Division (mathematics)
  • For example, the distinction between constitutive identity and a convenient observable transfers from Euclidean division gives integers q,r with a=bq+r and 0≤r<|b| for b≠0. to Polynomial long division over a field returns quotient and remainder of degree below the divisor, supporting gcd computations..
• Étale Algebra
  • A presentation A=k[x]/(f) is especially transparent when f is square-free: the derivative criterion \gcd(f,f')=1 detects separability

    This sourceFor the derivative criterion — f has no repeated root exactly when gcd(f, f') = 1, which is Lang's separability test for a polynomial; the étale-algebra reading of k[x]/(f) is the article's own framing. For the primitive-element theorem for finite separable extensions and for a separable element's minimal polynomial having no repeated root (Chapter V, algebraic extensions). For the input fact that R and C are the only finite extensions of R (Chapter XI, real fields); the R^r x C^s form for étale algebras is the article's own deduction from its earlier product theorem.

    SupportedVerified against the source

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

• Field (Algebraic)
  • Galois theory, which answers the question of when polynomial roots can be expressed in radicals, is entirely organised around the lattice of intermediate fields between a base field and a splitting field extension, with field automorphisms playing the role of symmetries

    This sourceLang's Algebra treating Galois theory through the correspondence between intermediate fields and the Galois group, with the criterion that an extension is solvable by radicals exactly when the associated group is solvable. Lang's Algebra defining perfect fields and separating characteristic-zero from characteristic-p behaviour, where in characteristic p the Frobenius identity (a + b)^p = a^p + b^p holds and the derivative of a polynomial can vanish.

    Supported in partVerified against a saved copy of the source

    “result of the Galois theory for finite Galois extensions”

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

• Formal derivative
  • It is recognized by its coefficient-and-exponent rule and the derivation identities it satisfies.
• Frobenius endomorphism
  • Characteristic p makes all intermediate binomial coefficients vanish, so the pth-power map preserves addition as well as multiplication and is natural under ring homomorphisms.
• Irreducible polynomial
  • An irreducible \(f\in F[x]\) generates a maximal ideal and the quotient \(F[x]/(f)\) is a field, making the property constructive in extension theory.
• Maschke's theorem
  • … the distinction between constitutive identity and a convenient observable transfers from Every complex representation of a finite group is a direct sum of irreducibles because complex characteristic is zero. to An algebraist checks modular characteristic first and uses counterexamples when char(k) divides |G|..
• Minimal polynomial (field theory)
  • The polynomial exists exactly for algebraic elements, is irreducible over the base field and generates the kernel of the evaluation homomorphism.
• Module (Algebra)
  • Standard algebra references use modules as the common language for ideals, abelian groups, linear representations, and homological constructions.
• Prime ideal
  • … in P has at least one factor in P. The scope is broad within that domain but bounded by the need for P is proper and every product lying in P has at least one factor in P. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Rational dependence
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Ring Homomorphism
  • Complex conjugation is an automorphism of \(\mathbb C\).
• Rupture field
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Splitting field
  • The move from The splitting field of \(x^3-2\) over \(\mathbb Q\) is \(\mathbb Q(\sqrt[3]{2},\zeta_3)\), not merely \(\mathbb Q(\sqrt[3]{2})\). to For \(x^q-x\) over \(\mathbb F_p\), where \(q=p^n\), the splitting field is \(\mathbb F_q\) because precisely its elements are the roots. demonstrates that continuity.
• Symmetric polynomial
  • For example, the distinction between constitutive identity and a convenient observable transfers from x²+y²+z²+xy+xz+yz is unchanged by every permutation of x,y,z. to Coefficients of a monic polynomial are expressed as elementary symmetric polynomials in its roots..
• Symmetrization
  • For two-variable bilinear maps over characteristic not two, symmetric and alternating parts are obtained by half the sum and half the difference.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 2 of 24 citations: 1 supported, 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

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