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Solution to Abstract Algebra by Dummit & Foote 3rd edition, Determine which matrices belong to a given set, Compute in a quotient of a polynomial ring, Basic properties of quotients of a polynomial ring, The complete homomorphic preimage of a prime ideal is a prime ideal, The product of two finitely generated ideals is finitely generated, If a prime ideal contains a product of ideals then it contains one of the factors, If a nontrivial prime ideal contains no zero divisors, then the ring is an integral domain, In an integral domain, two principal ideals are equal precisely when their generators are associates, In a polynomial ring, the ideal generated by the indeterminate is prime precisely when the coefficient ring is an integral domain, A ring has only the trivial one-sided ideals precisely when it is a division ring, Solution to Measure, Integration & Real Analysis by Axler, Solution to Mathematics for Machine Learning, The group of units of a field acts on the field by left multiplication, §2.2: Centralizers and Normalizers, Stabilizers and Kernels, §2.4: Subgroups Generated by Subsets of a Group, §2.5: The Lattice of Subgroups of a Group, §3.2: More on Cosets and Lagrange’s Theorem, §3.4: Composition Series and the Hölder Program, §3.5: Transpositions and the Alternating Group, §4.1: Group Actions and Permutation Representations, §4.2: Groups Acting on Themselves by Left Multiplication – Cayley’s Theorem, §4.3: Groups Acting on Themselves by Conjugation – The Class Equation, §5.2: The Fundamental Theorem of Finitely Generated Abelian Groups, §6.1: p-Groups, Nilpotent Groups, and Solvable Groups, §6.2: Applications in Groups of Medium Order, §7.2: Examples: Polynomial Rings, Matrix Rings, and Group Rings, §7.3: Ring Homomorphisms and Quotient Rings, §9.3: Polynomial Rings that are Unique Factorization Domains, §9.6: Polynomials in Several Variables over a Field and Gröbner Bases, §10.2: Quotient Modules and Module Homomorphisms, §10.3: Generation of Modules, Direct Sums, and Free Modules, §10.5: Exact Sequences – Projective, Injective, and Flat Modules, §11.2: The Matrix of a Linear Transformation, §11.5: Tensor Algebras, Symmetric and Exterior Algebras, §13.3: Classical Straightedge and Compass Constructions, §13.4: Splitting Fields and Algebraic Closures, §13.5: Separable and Inseparabel Extensions, §13.6: Cyclotomic Polynomials and Extensions, §14.2: The Fundamental Theorem of Galois Theory, §14.4: Composite Extensions and Simple Extensions, §14.5: Cyclotomic Extensions and Abelian Extensions over \mathbb{Q}, §14.7: Solvable and Radical Extensions: Insolvability of the Quintic, §14.8: Computation of Galois Groups over \mathbb{Q}, §14.9: Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups, §15.1: Noetherian Rings and Affine Algebraic Sets, §15.3: Integral Extensions and Hilbert’s Nullstellensatz, §17.1: Introduction to Homological Algebra – Ext and Tor, §17.3: Crossed Homomorphisms and H^1(G,A), §17.4: Group Extensions, Factor Sets, and H^2(G,A), §18.1: Linear Actions and Modules over Group Rings, §18.2: Wedderburn’s Theorem and Some Consequences, §18.3: Character Theory and the Orthogonality Relations, §19.1: Characters of Groups of Small Order, §19.3: Introduction to the Theory of Induced Characters. 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