Abstract Algebra
Explore groups and symmetry; rings and fields; extensions and actions. Concepts and connections are available independently of video coverage.
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Group actions
A group action assigns compatible transformations of a set to group elements.
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Burnside's lemma
For a finite group action, orbit count equals the average number of fixed points.
Combinatorial counting
Count finite outcomes with disjoint-case sums, successive-choice products and equal-fiber division or bijections. Check that cases are disjoint and multiplicities are constant. Multiplication of choice counts is not a probability-independence assumption.
Field extensions
A field extension contains a smaller field and can be viewed as its vector space.
Galois theory
Galois theory relates suitable field extensions to groups of automorphisms fixing the base field.
Integer partitions
A partition writes a positive integer as an unordered sum of positive integers.
Knowledge connections
- Galois theory · Related · Map context
- Field extensions · Related · Map context
- Combinatorial counting · Related · Map context
- Integer partitions · Related · Map context
- Burnside's lemma · Next concept · Map context
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Combinatorics
Explore basic counting; counting structures; partitions and symmetry. Concepts and connections are available independently of video coverage.