Engineering↗
Cyclic Extensions, Kummer Extensions and Solvability by Radicals
Cyclic Extensions, Kummer Extensions and Solvability by Radicals: core definitions, structural results and verification methods in abstract algebra.
Tag
extensions
Every page in the KEVOS library tagged extensions. 17 pages.
Engineering↗
Cyclotomic Fields and Cubic Galois Groups
Handbook guide to cyclotomic fields and cubic galois groups with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Field Extensions and Automorphism Correspondence
Handbook guide to field extensions and automorphism correspondence, covering definitions, key results, methods, verification checks and worked examples.
Engineering↗
Field Extensions, Minimal Polynomials and Extension Degree
Field Extensions, Minimal Polynomials and Extension Degree: core definitions, structural results and verification methods in abstract algebra.
Engineering↗
Fields: Axioms, Subfields, Extensions and Isomorphisms
A field is a set with addition and multiplication satisfying the familiar arithmetic laws, including additive inverses and multiplicative inverses for every nonzero element. Thi…
Engineering↗
Integral Extensions and Quadratic Extensions
Handbook guide to integral extensions and quadratic extensions with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Integral Extensions, Algebraic Integers and One-Dimensional Domains
Handbook guide to integral extensions, algebraic integers, with definitions, key results, methods, verification checks and worked examples.
Engineering↗
Semidirect Products, Extensions and Cohomological Classification
Handbook guide to semidirect products, extensions, with definitions, key results, methods, verification checks and worked examples.
Engineering↗
Separable and Normal Field Extensions
Handbook guide to separable and normal field extensions with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Transcendental Extensions and Transcendence Bases
Handbook guide to transcendental extensions and transcendence bases with core definitions, structural results, reasoning methods and verification checks.
Engineering↗
Computing Ext Groups
Techniques for computing Ext: choice of variable, use of the long exact sequences, standard computations over the integers, and Ext for cyclic and finitely generated modules.
Engineering↗
Extensions of Modules and the Baer Sum
Extensions of modules, equivalence of extensions, the Baer sum defined by pullback and pushout, and the resulting abelian group structure on Ext.
Engineering↗
The Ext Functor
Ext defined via extensions, via projective resolutions and via injective resolutions, the agreement of the three definitions, and the basic vanishing criteria.
Engineering↗
The Stein–Serre Theorem for Abelian Groups
The Stein-Serre theorem characterising countable torsion-free abelian groups that are free, via vanishing of Ext, and related Whitehead-type problems.
Engineering↗
The Tensor Product of Modules
The tensor product by universal property and construction, right exactness, flat modules, and the tensor-hom adjunction.
Engineering↗
The Tor Functor
Tor defined by projective resolutions, symmetry in its two variables, vanishing criteria, and the standard computations over the integers.
Engineering↗
The Two Long Exact Sequences of Ext
The long exact sequences of Ext in the first and second variables, their connecting homomorphisms, naturality, and the direction reversal caused by contravariance.
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