Courses / Mathematics / Abstract algebra II: rings, fields and Galois foundations Mathematics · unit version 1 Abstract algebra II: rings, fields and Galois foundations Proof-centered lessons from ideals and factorization through finite fields, Galois correspondence, solvability by radicals, modules, Smith classification, rational canonical forms, Noetherian ideal theory, discriminant methods and quadratic integrality, Dedekind factorization, ideal-class bounds and arithmetic Galois cycle certificates. Independent proof assessment and review remain necessary; this unit does not establish universal semester-course equivalence.
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What must a homomorphism do to the identity and operation, and why inspect its kernel? Ring maps add a multiplication contract to the familiar additive-group structure.
Compare reasoning and find a refresher A group homomorphism preserves the operation and hence the identity. Its kernel measures failure of injectivity and is a normal subgroup; a ring homomorphism must also preserve multiplication and, in this unit, 1.
Review homomorphisms and kernels
Why must a quotient operation be independent of representatives? Quotient rings require compatibility with both addition and multiplication.
Compare reasoning and find a refresher Different representatives name the same class, so replacing them must not change the output class. In rings an ideal provides the absorption property needed for multiplication.
Review quotient construction
What is the remainder when x²+1 is divided by x+1 over the rational numbers? Polynomial division connects evaluation, factors and extension fields.
Compare reasoning and find a refresher The remainder is 2, since evaluation at −1 gives 2. Over a field of characteristic 2 it becomes 0, illustrating why the coefficient field must be stated.
Review polynomial division
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Rings and fields: quotient, factorization and extension arguments
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One presentation, three scalar settings: an algebra manuscript
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Cumulative abstract-algebra practice examination
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Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Test ring axioms, units and zero divisors under explicit conventions.
Verify ring examples and nonexamples. Identify units and zero divisors. Prove finite-domain and cancellation results. 6 practice tasks.
Lesson details and objectives Prove quotient multiplication is well-defined exactly for ideals.
Check ideal absorption and additive closure. Prove quotient operations are independent of representatives. Identify proper and principal ideals. 6 practice tasks.
Lesson details and objectives Track identity preservation and quotient isomorphisms.
Verify ring homomorphisms. Prove the kernel is an ideal. Prove the first ring isomorphism theorem. 6 practice tasks.
Lesson details and objectives Prove the domain and field characterizations.
Prove the prime-ideal quotient criterion. Prove the maximal-ideal quotient criterion. Distinguish prime from maximal. 6 practice tasks.
Lesson details and objectives Prove Euclidean domains are principal and produce Bézout coefficients.
State a Euclidean division function. Derive principal ideals from a minimal remainder. Compute and verify a Bézout identity. 6 practice tasks.
Lesson details and objectives Use ideal chains and prime divisibility in a PID.
Prove irreducibles are prime in a PID. Prove factorization terminates. Distinguish Euclidean, PID and UFD implications. 6 practice tasks.
Lesson details and objectives Use leading coefficients, remainders and degree bounds.
Perform division over a field. Prove the factor and root-bound theorems. Explain why division can fail over a general ring. 6 practice tasks.
Lesson details and objectives Prove primitive products and track content.
Define content and primitive polynomials up to units. Prove Gauss’s lemma by reduction modulo primes. Transfer irreducibility to the fraction field. 6 practice tasks.
Lesson details and objectives Use rational roots, modular reduction and Eisenstein.
Apply degree-two/three root tests over fields. Prove and apply Eisenstein’s criterion. Distinguish sufficient tests from converses. 6 practice tasks.
Lesson details and objectives Construct the map and prove its kernel and surjectivity.
Prove the two-ideal CRT. Compute an explicit simultaneous residue. Distinguish comaximality from mere distinctness. 6 practice tasks.
Lesson details and objectives Build Frac(R) from a domain and prove its universal property.
Prove the fraction relation and operations are well-defined. Embed a domain into its fraction field. Extend injective maps into fields uniquely. 6 practice tasks.
Lesson details and objectives Separate algebraic and transcendental elements.
Define extension degree using a basis. Compute a simple quadratic basis. Distinguish F[α] from F(α). 6 practice tasks.
Lesson details and objectives Prove irreducibility, divisibility and basis consequences.
Construct and prove uniqueness of a minimal polynomial. Identify F[α] with F[x]/(mα). Compute multiplication and inverses in a quotient basis. 6 practice tasks.
Lesson details and objectives Use degree divisibility without assuming independence.
Prove the tower law. Use degree obstructions for containment. Find bases for a two-step extension. 6 practice tasks.
Lesson details and objectives Prove a degree obstruction and state its limits.
Derive quadratic field steps from line/circle intersections. Apply degree obstructions to classical constructions. Distinguish necessity from sufficiency of power-of-two degree. 11 practice tasks.
Lesson details and objectives Check arithmetic and prove finite field sizes.
Construct a small finite field by quotient. Prove finite fields have prime-power cardinality. Distinguish a field of size pⁿ from Z/pⁿZ. 6 practice tasks.
Lesson details and objectives Prove Frobenius and count its fixed points.
Prove the Frobenius homomorphism. Determine finite-field automorphism order. Describe subfields by divisibility. 6 practice tasks.
Lesson details and objectives Separate a root field from a splitting field.
Construct a splitting field by successive adjunction. Prove finiteness for a polynomial. Explain uniqueness through extension of embeddings. 6 practice tasks.
Lesson details and objectives Distinguish perfect fields from characteristic-p counterexamples.
Prove the gcd derivative criterion. Prove characteristic-zero and finite fields are perfect. Construct an inseparable extension. 6 practice tasks.
Lesson details and objectives Connect splitting fields, normality and automorphism bounds.
Define finite normal extensions and use the splitting-field criterion. Bound automorphisms by extension degree. Diagnose nonnormal and inseparable examples. 8 practice tasks.
Lesson details and objectives State finite Galois hypotheses and identify proof dependencies.
Apply the finite Galois correspondence with hypotheses. Relate subgroup indices to field degrees. Distinguish normal subgroups from arbitrary subgroups. 8 practice tasks.
Lesson details and objectives Compute biquadratic and complete S₃ correspondences, distinguishing fixed fields from normal extensions.
Compute automorphisms and fixed fields explicitly. Check the subgroup-field diagram by degrees. Identify what introductory Galois results do not establish. 8 practice tasks.
Lesson details and objectives Prove the finite-separable primitive-element theorem and test its assumptions.
Construct and verify a primitive element. Prove the theorem for infinite and finite base fields. Exhibit a finite extension that is not simple when separability fails. 6 practice tasks.
Lesson details and objectives Prove both directions using cyclic resolvents and solvable Galois envelopes.
Prove the cyclic radical-extension lemmas. Construct solvable Galois envelopes for radical towers. Derive the characteristic-zero solvability criterion and apply its obstruction. 6 practice tasks.
Lesson details and objectives Build modules, quotients and operator actions with explicit scalar-ring hypotheses.
Prove module quotient and isomorphism results. Relate abelian groups and operators to modules. Distinguish generation, freeness, torsion and splitting. 6 practice tasks.
Lesson details and objectives Prove reversible presentation changes and finite generation of integer subgroups.
Interpret column relations and cokernels. Prove unimodular operations preserve the quotient. Prove subgroups of Zⁿ are free and derive finite presentations. 6 practice tasks.
Lesson details and objectives Prove termination and divisibility in Smith reduction while tracking exact transformations.
Track unimodular row and column operations. Prove Euclidean reduction and pivot repair terminate. Handle rectangular and zero matrices. 6 practice tasks.
Lesson details and objectives Prove uniqueness and classification, then solve integer systems completely.
Derive invariant factors from determinantal divisors. Prove finitely generated abelian classification and intrinsic uniqueness. Certify integer consistency, kernels and complete solution families. 6 practice tasks.
Lesson details and objectives Extend Smith reduction to field polynomials and prove monic invariant-factor uniqueness.
Prove polynomial unimodularity and allowed operations. Prove degree-decreasing reduction and divisibility repair. Recover monic factors from determinantal divisors. 6 practice tasks.
Lesson details and objectives Prove the evaluation quotient and connect polynomial equivalence to constant similarity.
Prove the evaluation map and its full kernel by telescoping. Explain the finite-dimensional quotient of a full-rank polynomial matrix. Translate module isomorphisms into constant intertwining maps. 6 practice tasks.
Lesson details and objectives Construct rational canonical form over any field without assuming eigenvalues split.
Build companion bases from cyclic polynomial modules. Prove invariant-factor similarity classification and polynomial formulas. Extract and verify an actual cyclic basis and compare nonsimilar examples. 6 practice tasks.
Lesson details and objectives Prove ideal correspondence and preserve the hypotheses in quotient calculations.
Prove the inverse ideal maps. Transport prime and maximal ideals. Compute ideal operations and iterated quotients. 6 practice tasks.
Lesson details and objectives Use norms and a nonprincipal ideal to test integer-like claims in a quadratic ring.
Prove irreducibility and nonassociation by norms. Separate prime from irreducible. Compute a nonprincipal ideal and its square. 6 practice tasks.
Lesson details and objectives Connect ascending chains, finite ideal generators and Hilbert’s basis theorem.
Prove the two Noetherian criteria. Construct finite polynomial-ideal generators by degree reduction. Apply quotient inheritance and distinguish PID, UFD and Noetherian. 6 practice tasks.
Lesson details and objectives Construct elementary-symmetric expressions and derive Newton identities without hidden division.
Prove existence and uniqueness of symmetric expressions. Derive and use Newton identities. Diagnose positive-characteristic failures of coefficient recovery. 6 practice tasks.
Lesson details and objectives Connect squared root differences, derivative products and a quotient-algebra determinant.
Prove the discriminant identities. Derive low-degree formulas from coefficients. Compare discriminants with derivative gcds across characteristics. 6 practice tasks.
Lesson details and objectives Prove the sign criterion and classify irreducible cubic examples with explicit boundaries.
Prove the Vandermonde sign action. Identify the quadratic field fixed by the sign kernel. Classify cubic groups after checking irreducibility and characteristic. 6 practice tasks.
Lesson details and objectives Prove integrality criteria and derive the correct quadratic integral basis.
Prove finite-module criteria, closure and transitivity. Derive the quadratic integer ring including denominator restrictions. Distinguish a quadratic order from the full integrally closed ring. 6 practice tasks.
Lesson details and objectives Connect multiplication matrices, ideal membership and finite quotient indices.
Prove nonzero ideals have full lattice rank and finite index. Calculate principal indices from field norms. Prove nonzero prime ideals are maximal without assuming factorization. 6 practice tasks.
Lesson details and objectives Use exact products to prove invertibility and expose failure in a nonmaximal order.
Construct the group of invertible fractional ideals. Prove inverse and cancellation criteria. Certify an ideal class of exact order two and a noninvertible boundary case. 6 practice tasks.
Lesson details and objectives Derive invertibility and prime-ideal factorization from the ring hypotheses.
Construct a prime ideal inverse without assuming it exists. Prove factorization by strict ascending enlargement and uniqueness by cancellation. Verify the hypotheses for full quadratic integer rings. 6 practice tasks.
Lesson details and objectives Connect invertible-ideal quotients to exact index formulas and prime powers.
Prove the one-dimensional quotient lemma. Derive multiplicativity from prime factorization and index chains. Distinguish equal quotient sizes from different ring structures. 6 practice tasks.
Lesson details and objectives Prove a planar lattice bound and exhaust the classes of integers adjoin square root of minus five.
Derive the convex-body lattice theorem from an area argument. Bound integral representatives in every ideal class. Prove the complete class group has exactly two elements. 6 practice tasks.
Lesson details and objectives Prove how irreducible factors partition finite-field roots into permutation cycles.
Prove the orbit polynomial is the minimal polynomial. Compute a checked orbit and squarefree factorization. Explain repeated-factor and degree-loss boundaries. 6 practice tasks.
Lesson details and objectives Construct a residue field and prove one automorphism lifts Frobenius on all roots.
Construct a finite residue field from the integral root algebra. Prove the formal orbit-polynomial lifting argument. Identify the exact role of squarefree reduction and the limits of the theorem. 6 practice tasks.
Lesson details and objectives Combine proved modular cycles with a subgroup argument to identify S4.
Certify the two finite factorizations and characteristic-zero irreducibility. Prove the subgroup exclusion rather than guessing from cycle types. Separate solvability by radicals from quadratic constructibility. 6 practice tasks.
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