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Abstract Algebra
Explore groups and symmetry; rings and fields; extensions and actions. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Bijections
A bijection is both injective and surjective and has a two-sided inverse.
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Groups and symmetry
Binary operations
A binary operation combines two elements into one in a specified set.
Video explanations to be addedGroups
A group has an associative operation, an identity and an inverse for every element.
Video explanations to be addedSubgroups
A subgroup is a subset that forms a group under the inherited operation.
Video explanations to be addedCyclic groups
A cyclic group is generated by repeated powers of a single element.
Video explanations to be addedCosets and Lagrange's theorem
Cosets partition a finite group, so a subgroup's order divides the group's order.
Video explanations to be addedGroup homomorphisms
A homomorphism preserves group multiplication; its kernel measures failure of injectivity.
Video explanations to be addedQuotient groups
Cosets of a normal subgroup inherit a well-defined group operation.
Video explanations to be addedNormal subgroups
A subgroup invariant under conjugation allows consistent multiplication of cosets.
Video explanations to be addedSylow theorems
For finite groups, Sylow theorems constrain existence, conjugacy and number of maximal prime-power subgroups.
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Rings and fields
Rings
A ring supports addition and multiplication linked by distributive laws.
Video explanations to be addedIdeals
An ideal is an additive subgroup closed under multiplication by ring elements on the required sides.
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A field is a commutative ring where every nonzero element has a multiplicative inverse.
Video explanations to be addedPolynomial rings
Polynomials over a ring form a ring with coefficientwise addition and polynomial multiplication.
Video explanations to be addedQuotient rings
Reducing ring elements modulo an ideal gives a well-defined quotient ring.
Video explanations to be addedIntegral domains
An integral domain is a nonzero commutative ring with identity and no nonzero zero divisors.
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Every ideal in a principal ideal domain is generated by one element.
Video explanations to be addedUnique factorization domains
Nonzero nonunits factor into irreducibles uniquely up to order and units.
Video explanations to be addedModules
Modules generalize vector spaces by allowing scalars from a ring rather than a field.
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Extensions and actions
Field extensions
A field extension contains a smaller field and can be viewed as its vector space.
Video explanations to be addedGalois theory
Galois theory relates suitable field extensions to groups of automorphisms fixing the base field.
Video explanations to be addedGroup actions
A group action assigns compatible transformations of a set to group elements.
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