Abstract Algebra II
Skills
Every skill in the course, and how you are doing on each.
Rings, ideals, quotient rings, and homomorphisms
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Subring test
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Modular arithmetic
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Zero divisors
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Residue rings
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Idempotents
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Integer ideals
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Principal ideals
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Ideal intersection
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Integer ring
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Prime divisibility
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Bezout identity
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Evaluation kernel
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Factor theorem
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Quotient map
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Coset equality
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Ideal correspondence
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Honors extension
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Field quotient
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Chinese remainder theorem
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Comaximal ideals
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Divisor lattice
115 more topics with one problem
- Integer arithmetic
- Multiplicative identity
- Ring axioms
- Field axioms
- Multiplicative inverses
- Ring structure
- Complex multiplication
- Gaussian integers
- Even integers
- Identity hypothesis
- Subring conventions
- Minimal positive integer
- Residue classes
- Ring characteristic
- Coprimality
- Modular inverses
- Units modulo n
- Annihilators
- Nilpotence index
- Nilpotent elements
- Congruence solving
- Coordinatewise operations
- Direct product rings
- Idempotent elements
- Divisibility
- Ideal test
- Ideal membership
- Integer divisibility
- Absorption
- Closure properties
- Generated ideals
- Ideal sum
- Minimality
- Ideal containment
- Ideal product
- Strict inclusion
- Ideal absorption
- Proper ideals
- Unit elements
- Field ideals
- Ideal classification
- Nonzero elements
- Prime ideals
- Maximal ideals
- Gcd ideals
- Integer ideal lattice
- Lcm ideals
- Additive multiples
- Integer universal map
- Unital homomorphisms
- Homomorphism kernel
- Two sided ideals
- Zero fiber
- Homomorphism image
- Unital maps
- Polynomial evaluation
- Ring homomorphisms
- Substitution
- Principal polynomial ideal
- First isomorphism theorem
- Integer kernel
- Kernel computation
- Product projection
- Quotient isomorphism
- Ambient absorption
- Ideal image counterexample
- Nonsurjective map
- Difference argument
- Injective homomorphism
- Trivial kernel
- Factor homomorphism
- Quotient universal property
- Well definedness
- Quotient ring operations
- Representative independence
- Two sided ideal
- Modular equations
- Quotient arithmetic
- Residue inverses
- Ideal cosets
- Modular congruence
- Preimage image bijection
- Iterated quotients
- Third isomorphism theorem
- Maximal ideal criterion
- Domain quotient
- Prime ideal criterion
- Zero products
- Radical ideals
- Reduced quotients
- Explicit isomorphism
- Simultaneous congruences
- Chinese remainder isomorphism
- Intersection product equality
- Matrix rings
- Noncommutative example
- One sided ideals
- One sided ideal
- Quotient ring obstruction
- Representative dependence
- Matrix ring center
- Scalar matrices
- Definition audit
- Proper ideal condition
- Zero ring
- Comparability
- Ideal union
- Identity preservation
- Nonunital homomorphism
- Coordinate idempotents
- Product ring ideals
- Idempotent splitting
- Product decomposition
- Prime ideal counterexample
- Residue ring ideals
Domains, divisibility, and factorization
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Integral domain
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Associates
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Irreducible elements
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Euclidean algorithm
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Integer gcd
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Two generated ideal
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Factorization uniqueness
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Factorization hierarchy
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Leading coefficient
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Quartic irreducibility
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Prime exponents
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Fraction field
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Primitive polynomial
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Rational root theorem
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Nonprincipal ideal
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Two variable polynomials
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Polynomial ufd
112 more topics with one problem
- Zero product property
- Domain cancellation
- Factorization
- Zero divisor free
- Divisibility relation
- Preorder
- Divides one
- Unit characterization
- Universal divisors
- Associate elements
- Principal ideal equality
- Unit multiples
- Associate classes
- Common divisors
- Gcd uniqueness
- Associate normalization
- Integer units
- Domain factorization
- Prime elements
- Irreducible not prime
- Norm argument
- Quadratic integer ring
- Bezout certificate
- Euclidean division
- Negative integers
- Remainder uniqueness
- Coprime polynomials
- Polynomial euclidean algorithm
- Successive remainders
- Bezout coefficients
- Extended euclidean algorithm
- Gcd generator
- Factorization identity
- Monic divisor
- Polynomial division
- Euclidean domain
- Minimal euclidean value
- Principal ideal domain
- Membership divisibility
- PID ideals
- Principal generator
- Irreducible to prime
- PID factorization
- Principal ideal sum
- Bezout domain
- PID gcd
- Maximality
- PID prime ideals
- Principal containment
- Ascending chain condition
- Divisibility reversal
- Nonunit factor
- Principal ideal chain
- Proper factorization
- Acc principal ideals
- Factorization existence
- Termination argument
- Ideal chain termination
- Irreducible products
- PID factorization existence
- Associate factors
- Existence and uniqueness
- PID implies ufd
- Monic factorization
- Polynomial associates
- Eisenstein after translation
- Real factorization
- Coefficient content
- Primitive polynomials
- Gauss transfer
- Primitive cubic
- Rational factorization
- Integer factorization
- Ufd gcd
- Gcd lcm identity
- Ufd lcm
- Euclid lemma
- Fraction uniqueness
- Reduced fractions
- Equivalence relation
- Well defined operations
- Domain preservation
- Localization
- Coprime numerator denominator
- Gaussian norm
- Gaussian units
- Multiplicative norm
- Gaussian euclidean domain
- Nearest lattice point
- Complex division
- Euclidean remainder
- Gaussian gcd
- Fraction field universal property
- Induced field map
- Injective embedding
- Ufd not PID
- Nonunique factorization
- Norm comparison
- Quadratic domain
- Pell type growth
- Quadratic ring units
- Nonprime principal ideal
- Quadratic integers
- Quotient zero divisor
- Gaussian ufd
- Localization of ufd
- Prime factor inversion
- Content factorization
- Hypothesis audit
- Irreducible prime gap
- Gcd not bezout
- Classification synthesis
Polynomial rings and irreducibility
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Gauss lemma
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Linear factor
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Irreducible factors
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Formal derivative
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Repeated roots
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Finite fields
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Eisenstein criterion
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Shifted eisenstein
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Irreducible polynomial
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Quartic factorization
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Reciprocal polynomial
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Characteristic p
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Zero derivative
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Degree bound
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Correspondence theorem
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Primary components
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Artin schreier
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Prime field
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Mobius counting
112 more topics with one problem
- Degree additivity
- Leading coefficients
- Constant polynomials
- Degree
- Polynomial units
- Evaluation map
- Remainder theorem
- Synthetic division
- Linear divisibility
- Roots
- Distinct roots
- Root bound
- Zero polynomial
- Polynomial content
- Primitive part
- Cubic factorization
- Irreducibility test
- Counterexample
- Modular reduction
- One way criterion
- Low degree test
- Quartic counterexample
- Field dependent factorization
- Fourth powers
- Polynomial gcd
- Multiplicity
- Fermat polynomial
- Simple roots
- Evaluation quotient
- First isomorphism
- Linear ideal
- Characteristic edge case
- Quadratic roots
- Small finite fields
- Rational irreducibility
- Translation automorphism
- Finite field gcd
- Quintic reduction
- Clearing denominators
- Primitive factors
- Maximal ideal
- Generated ideal
- Monic gcd
- Polynomial PID
- Evaluation pairs
- Polynomial crt
- Root test limit
- Sophie germain
- Constant term
- Factor reversal
- Finite field enumeration
- Irreducible quadratics
- Root testing
- Explicit inverse
- Finite field quotient
- Normal forms
- Derivative gcd
- Squarefree polynomials
- Frobenius exponents
- Cyclotomic factors
- Quadratic irreducibility
- Sixth roots
- Lagrange interpolation
- Uniqueness
- Binomial coefficients
- Prime cyclotomic
- Degree drop
- Reduction hypotheses
- Binary quartic
- Finite field irreducibility
- Quadratic factor test
- Degree four
- Finite field factors
- Frobenius gcd test
- Quotient ideal classification
- Repeated factor crt
- Degree four extension
- Nested radicals
- Eisenstein quotient
- Field basis
- Inverse from relation
- Root translates
- Coefficient comparison
- Even quartic
- Quadratic factorization
- Coefficient reversal
- Irreducible quintic
- Reciprocal eisenstein
- Binary sextic
- Irreducibility certificate
- Low degree divisors
- Factor degree sieve
- Modular lifting
- Multiple primes
- Binomial parity
- Degree eight irreducibility
- Power of two
- Binary quartics
- Irreducible count
- Irreducible prime
- Perfect fields
- Pth root recursion
- Squarefree factorization
- Nilpotent cosets
- Reduced quotient
- Squarefree modulus
- Finite field master factorization
- Frobenius fixed points
- Irreducible degrees
- Cross characteristic
- Factorization capstone
- Quotient structure
Modules and finitely generated modules over a PID
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Linear dependence
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Invariant factors
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Elementary divisors
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Primary decomposition
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Torsion free quotient
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Mobius inversion
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Scalar extension
130 more topics with one problem
- Integer modules
- Scalar action
- Submodule classification
- Well ordering
- Annihilator
- Cyclic modules
- Element orders
- Lattice submodules
- Quotient modules
- Direct sums
- Module generators
- Noncyclic groups
- Image submodule
- Kernel submodule
- Linear maps
- Cyclic quotients
- Ideal lattice
- Annihilator product
- Torsion submodule
- Annihilator ideal
- Empty subset edge case
- Universal condition
- Free modules
- Integer basis
- Unimodular matrix
- Integer span
- Redundant generators
- Determinant obstruction
- Integer linear system
- Lattice membership
- Free part
- Quotient decomposition
- Torsion coordinate
- Cokernel
- Diagonal map
- Finite index
- Cokernel isomorphism
- Elementary operations
- Presentation matrices
- Cokernel classification
- Determinantal divisors
- Smith normal form
- Minor gcd
- Smith invariants
- Three generator presentation
- Generators relations
- Presentation matrix
- Invariant factor conversion
- Finite abelian groups
- Partition enumeration
- Element exponent
- Isomorphism invariants
- Premise checking
- Free rank
- Structure theorem
- Finite generation
- Splitting
- Module rank
- Tensor localization
- Coordinate projection
- PID induction
- Submodule freeness
- Finite torsion modules
- Hypothesis boundary
- Residue fields
- Non PID counterexample
- Rank one submodule
- Finite generation edge case
- Rational module
- Torsion free nonfree
- Chinese remainder module
- Coprime ideals
- Gcd count
- Generator image
- Hom cyclic modules
- Cyclic automorphisms
- Endomorphism ring
- Unit group
- Coprimality criterion
- Cyclic direct sum
- Element order
- Minimal generators
- Mod p quotient
- Nakayama style bound
- Element order count
- Finite modules
- Free cokernel
- Kernel rank
- Rank deficient smith form
- Congruence obstruction
- Diophantine system
- Smith solvability
- Direct sum decomposition
- Free module lifting
- Split exact sequence
- Cyclic quotient
- Nonsplit extension
- Torsion obstruction
- Bezout functional
- Direct summand
- Primitive vector
- Direct summand criterion
- Saturated submodule
- Invariant subspace
- Linear operator
- Polynomial module
- Annihilator polynomial
- Companion matrix
- Cyclic vector
- Jordan blocks
- Kernel growth
- Nilpotent invariant factors
- Companion blocks
- Field extension
- Rational canonical form
- Module classification
- Similarity counterexample
- Geometric multiplicity
- Jordan form
- Diagonalizability criterion
- Edge cases
- Squarefree minimal polynomial
- Field dependent diagonalization
- Rotation operator
- Block reconstruction
- Kernel growth inversion
- Nilpotent classification
- Hom count
- Presentation capstone
- Smith determinantal divisors
Algebraic field extensions and minimal polynomials
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Minimal polynomial
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Quadratic extension
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Power basis
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Intermediate field
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Tower law
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Intermediate fields
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Biquadratic field
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Compositum degree
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Product basis
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Algebraic tower
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Simple extension
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Relative minimal polynomial
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Cyclotomic extension
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Conjugate roots
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Field embeddings
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Inseparable degree
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Mixed extension
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Separable degree
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Coprime degrees
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Field norm
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Field trace
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Primitive element theorem
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Purely inseparable
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Frobenius orbit
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Power basis discriminant
93 more topics with one problem
- Coordinate uniqueness
- Extension basis
- Algebraic elements
- Rational function field
- Transcendental element
- Cubic extension
- Extension arithmetic
- Relation reduction
- Degree divisibility
- Cubic field
- Prime degree extension
- Algebraic closure operations
- Finite generated extension
- Finite implies algebraic
- Finite extension
- Coefficient field
- Finite subextension
- Polynomial translation
- Shifted generator
- Cubic quotient
- Finite field cardinality
- Vector space count
- Base field change
- Degree six polynomial
- Mixed radicals
- Primitive generator
- Eisenstein extension
- Field lattice
- Prime degree
- Tower basis
- Real subfield
- Finite field compositum
- Finite field intersection
- Gcd lcm degrees
- Real embedding
- Separability
- Frobenius map
- Imperfect field
- Perfect field
- Normal extension
- Splitting criterion
- Embedding invariance
- Galois subextension
- Normal intersection
- Biquadratic embeddings
- Normal separable
- Sign conjugates
- Compositum
- Trivial intersection
- Split quadratic
- Tensor algebra
- Multiplication matrix
- Inverse algebraic element
- Simple field generator
- Bad parameters
- Embedding separation
- Intermediate base
- Minimal polynomial change
- Embedding count
- Nonsimple extension
- Primitive element failure
- Base change factorization
- Cubic self tensor
- Normal compositum
- Separable compositum
- Splitting fields
- Coprime intersection
- Quintic splitting field
- Conjugates
- Finite field minimal polynomial
- Finite field counting
- Proper subfields
- Quartic irreducibles
- Minimal polynomial derivative
- Biquadratic tower
- Norm transitivity
- Trace transitivity
- Finite separable extension
- Minimal polynomial computation
- Power sum recurrence
- Real cyclotomic subfield
- Geometric nonreducedness
- Inseparable tensor product
- Nilpotent base change
- Cubic norm
- Cubic trace
- Finite field capstone
- Trace norm
- Cubic splitting field
- Tensor compositum
- Biquadratic lattice
- Fixed sign symmetry
- Transcendence basis
Finite Galois theory
122 more topics with one problem
- Conjugation automorphism
- Galois definition
- Cubic conjugates
- Degree computation
- Frobenius automorphism
- Complex conjugation
- Fixed field
- Quadratic galois
- Root action
- Small splitting field
- Pure cubic
- Splitting field degree
- Symmetric group three
- Frobenius generator
- Irreducible cubic
- Inseparable extension
- Galois hypotheses
- Nonnormal cubic
- Separable extension
- Characteristic boundary
- Fixed elements
- Quadratic galois general
- Imaginary quadratic
- Root of unity
- Third cyclotomic field
- Biquadratic galois group
- Fixed field computation
- Klein four
- Galois nontransitivity
- Normality counterexample
- Quartic radical
- Biquadratic generator
- Primitive element
- Fundamental theorem galois
- Inclusion reversal
- Normal subgroup
- Cubic subfields
- Normal subgroups
- Field lattice
- Dihedral galois group
- Explicit automorphisms
- Quartic splitting field
- Alternating subgroup
- Discriminant square
- Vandermonde
- Cubic discriminant
- Cyclic cubic
- Finite field subfields
- Frobenius subgroups
- Field sixteen elements
- Irreducible quartic
- Primitive element finite field
- Quadratic unit
- Finite field norm
- Finite field trace
- Frobenius conjugates
- Eighth cyclotomic field
- Klein four group
- Quadratic subfields
- Maximal real subfield
- Seventh cyclotomic field
- Unique subfields
- Field intersection
- Linearly disjoint
- Product galois group
- Restriction map
- Cyclotomic tower
- Exact sequence
- Restriction homomorphism
- Constructible polygons
- Cyclotomic degree
- Two group tower
- Roots of unity
- Solvability by radicals
- Solvable group
- Cyclic quartic
- Fourth roots unity
- Kummer extension
- Artin fixed field
- Automorphism independence
- Normal basis
- Orbit degree
- Root stabilizer
- Galois correspondence
- Nonnormal quartic
- Dihedral fixed fields
- Quartic root field
- Subgroup normality
- Finite field factorization
- Irreducible polynomial count
- Dedekind cycle type
- Symmetric quintic
- Transposition generation
- Discriminant parity
- Quartic resolvent
- Criterion
- Gram determinant
- Separability criterion
- Trace pairing
- Intersection subgroup
- Restriction isomorphism
- Cubic resolvent
- Explicit s4 quartic
- Mod two irreducibility
- Conjugate fields
- Subgroup core
- Quotient galois group
- Restriction exact sequence
- Contrast
- Cyclic coboundary
- Hilbert ninety
- Norm one
- Cubic radical
- Cyclic eigenvector
- Kummer classification
- Cyclic degree p
- Fifteenth cyclotomic field
- Quadratic characters
- Subgroup lattice count
- Frobenius group order twenty
- Pure quintic capstone
- Semidirect product