Engineering articles and subject areas in the KEVOS knowledge library. 2176 pages.
Dense and sparse polynomial representation, multiplication algorithms, pseudo-division over a UFD, primitive parts and content, and the subresultant and modular remedies for coe…
The ring F[X] as a Euclidean domain, division with remainder, roots and degree bounds, irreducibility, unique factorization, Gauss's lemma for ℤ[X], and polynomial congruences a…
The order-theoretic and algebraic definitions of a lattice, the equivalence between them, and why universal algebra needs the algebraic form: only an equationally defined class …
Sets, proper classes, relations, kernels and the notational conventions universal algebra shares with model theory: why the collection of all groups cannot be a set, how relatio…
The preservation theorems: Łoś–Tarski for substructures, Lyndon for homomorphisms, Keisler–Galvin for reduced products, Chang–Łoś–Suszko for unions of chains, and their relation…
Primal algebras, Foster's theorem that they generate varieties equivalent to Boolean algebras, the representation of every member as a Boolean power, and the relationship to fun…
The separation between primality testing and factoring, compositeness tests versus primality proofs, probable primes, pseudoprimes, and how to select a testing strategy.
Principal congruence formulas, Mal'cev's chain characterisation, when the chains are uniformly bounded, and the application to bounding the size of subdirectly irreducible algeb…
Models of randomised computation, one-sided and two-sided error, error reduction by repetition, generating random numbers in a range without bias, rejection sampling, and genera…
Trial division, the Fermat test and its failure on Carmichael numbers, the Miller–Rabin test with its 1/4 error bound, practical prime generation, perfect power testing and the …
Universal properties, products and coproducts in a general category, initial and terminal objects, and uniqueness up to unique isomorphism.
Projective classes, relative projectives, allowable epimorphisms, relative exactness, and the examples that motivate relative homological algebra.
Structure of projective and finitely generated modules over a PID, the structure theorem, torsion and free parts, and the resulting short projective resolutions.
Pullbacks and pushouts, their construction in module categories, limits and colimits over a diagram, and exactness of filtered colimits.
Quadratic fields, their discriminants and integral bases, prime decomposition by the Kronecker symbol, binary quadratic forms, reduction, and the correspondence between form cla…
Quadratic residues modulo a prime and a composite, Euler's criterion, the Legendre and Jacobi symbols, the law of quadratic reciprocity and its supplements, and the quadratic re…
Quasigroups and loops as equationally defined algebras, the correspondence with Latin squares, why division must be in the type for quasigroups to form a variety, and the congru…
Quasiprimal algebras, Pixley's characterisation via inner isomorphisms, the relationship to primality, and the varieties quasiprimal algebras generate.
Regulators and fundamental units of real quadratic fields, the continued fraction algorithm, the exponential cost problem, and Shanks's infrastructure with the distance function.
The source's Recent Developments chapter: the commutator programme, the classification of varieties by congruence conditions, decidability questions, Boolean constructions, stru…
Reduced products over a filter, the congruence they induce, the direct product and ultraproduct as extreme cases, and the Horn sentence preservation theorem.
Construction of relative derived functors from relative resolutions, their properties, the comparison with absolute derived functors, and worked examples.
The standard, matrix, conjugate-vector and minimal-polynomial representations of an algebraic number, their relative costs, and how to convert between them.
The bar resolution and its normalised version, cocycles and coboundaries in explicit terms, resolutions from presentations, and the trade-off between explicitness and size.
Commutative rings with unity, units and zero divisors, integral domains and fields, ideals and quotient rings, ring homomorphisms and the isomorphism theorems, with the CRT as a…
Numerical root finding for polynomials over C: conditioning, the Newton and Aberth methods, splitting-circle approaches, and the precision management needed to support exact num…
Left and right satellites of an additive functor, their construction, the connection with derived functors, and their use when the category lacks enough projectives.
The satisfaction relation defined by recursion, truth for sentences, theories and models, elementary equivalence, and why elementarily equivalent structures can be non-isomorphic.
Semantic embeddings as a technique for transferring undecidability, the interpretation of graphs and semigroups into varieties, the decidability results that stand in contrast, …
The Killing form and Cartan's criterion, the Casimir element, the two Whitehead lemmas, and their consequences including Weyl's complete reducibility theorem and Levi's theorem.
Semisimple varieties, the relationship to discriminator varieties, residual smallness, and the structure available when subdirect decomposition terminates in simple algebras.
The SQUFOF algorithm: continued fraction expansion of the square root, square forms, the reverse cycle, multipliers, and why it excels in a narrow but important range.
Skew congruences in direct products, skew-free algebras and varieties, the relationship to congruence permutability and distributivity, and the role in Boolean representations.
Root-finding modulo a prime: the gcd with x^p − x, equal-degree splitting by random shifts, and the special low-degree cases that admit closed forms.
Computing modular square roots: the p ≡ 3 (mod 4) shortcut, the Tonelli–Shanks algorithm and its 2-adic structure, and Cornacchia's algorithm for x² + dy² = p.
Steiner triple systems recast as algebras: squags (Steiner quasigroups) and sloops (Steiner loops), the equational axioms for each, and what the variety structure delivers that …
Stone duality: the Stone space of ultrafilters, Boolean spaces as compact Hausdorff totally disconnected spaces, the dual equivalence of categories, and how it seeds the Boolean…
Subdirect products, subdirect irreducibility, the monolith, and Birkhoff's subdirect representation theorem with its Zorn's lemma proof.
Smooth number densities, the index calculus method for discrete logarithms, the quadratic sieve and number field sieve for factoring, practical improvements, and what record com…
Restriction and corestriction (transfer) maps, the index formula, consequences for torsion, and detection on Sylow subgroups.
Subuniverses, subalgebras, the generation operator Sg(X), and the two equivalent descriptions of generation as intersection from above and as term-closure from below.
The syntactic congruence and syntactic monoid, Kleene's characterisation of recognisable languages as the regular ones, Schützenberger's star-free theorem, and Eilenberg's varie…
Tame congruence theory: minimal sets, the five local types, type sets of varieties, and how the classification refines and largely supersedes the 1981 congruence-condition hiera…
Terms as syntactic objects, the term algebra T(X) and its absolute freeness, the distinction between a term and the operation it induces, and why the term algebra is the referen…
The constraint satisfaction problem, polymorphisms as an algebraic invariant, the Feder–Vardi dichotomy conjecture, the algebraic reformulation, and the 2017 resolution.
The Boolean Prime Ideal Theorem, its equivalent formulations, its strength relative to the axiom of choice, and which results in this collection depend on it.
The centre of an arbitrary algebra, its first-order definition, the characterisation of algebras with Z(A) = ∇ as polynomially equivalent to modules, and the commutator programm…
The Chevalley-Eilenberg resolution of the trivial module, its differential, the resulting cochain complex, and the bound on cohomological dimension.