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

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.

• 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.
• Univariate
  • … between constitutive identity and a convenient observable transfers from A polynomial in x with fixed real coefficients is univariate even when it contains many powers of x. to A report distinguishes a univariate outcome analysis from a model with one outcome but several predictors, whose terminology varies by field..

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