MATH-19 · Advanced · ENGLISH · PLANNED
Build the picture, explain the rule, work an example, then justify it.
Study time is an initial activity estimate, excluding extra reading, revision and extended laboratory or project work. Videos and question bodies have not been produced by adding this outline.
Complete these courses or demonstrate equivalent knowledge. These are recommended prerequisites, not enforced enrolment locks.
YouTube playlist: ARK Mathematics | Abstract Algebra and Number Theory
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Sign in to save this learning pathPractical outcome: Describe the symmetries of a manufactured tile.
Explain binary operations using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain group axioms using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain subgroups using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain cyclic groups using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain permutation groups using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Use the first concepts from this section. Describe the symmetries of a manufactured tile. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain cosets using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain lagrange's theorem using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain normal subgroups using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain quotient groups using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Explain group homomorphisms using a visual and one worked real-world example. Context: Describe the symmetries of a manufactured tile. End with a prediction or check-for-understanding.
Describe the symmetries of a manufactured tile. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Binary operations, Group axioms, Subgroups and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Explain polynomial arithmetic in a small finite field.
Explain rings using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain ideals using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain integral domains using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain polynomial rings using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Use the first concepts from this section. Explain polynomial arithmetic in a small finite field. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain unique factorisation using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain field extensions using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain finite fields using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain irreducible polynomials using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain galois correspondence introduction using a visual and one worked real-world example. Context: Explain polynomial arithmetic in a small finite field. End with a prediction or check-for-understanding.
Explain polynomial arithmetic in a small finite field. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Rings, Ideals, Integral domains and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Practical outcome: Analyse modular check digits for a stock-code system.
Explain euclidean algorithm using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain bezout's identity using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain prime factorisation using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain modular arithmetic using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Use the first concepts from this section. Analyse modular check digits for a stock-code system. Produce a labelled diagram, trace or calculation; explain each step and check one changed input.
3–6 questions on the preceding first-half concepts only: recall, visual interpretation, application and one plausible misconception. Explain each answer after submission.
Explain chinese remainder theorem using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain fermat's little theorem using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain euler's theorem using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain quadratic residues using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Explain diophantine equations using a visual and one worked real-world example. Context: Analyse modular check digits for a stock-code system. End with a prediction or check-for-understanding.
Analyse modular check digits for a stock-code system. Work without the example, compare two cases, record assumptions and limitations, then self-review correctness, reasoning and clarity.
6–12 questions covering Euclidean algorithm, Bezout's identity, Prime factorisation and the remaining section concepts, with cumulative retrieval from earlier sections. Include a new transfer problem; 70% unlocks progression.
Combine all three sections: Analyse modular check digits for a stock-code system. Submit an original worked artifact, evidence or tests, assumptions and a reflection. Rubric: correctness, method, practical usefulness, reproducibility and communication. Self-reviewed, not certified.
6–12 balanced questions sampling every section of Abstract Algebra and Number Theory, including interpretation, calculation or reasoning, a misconception and a new application. Passing is 70%; review errors before retrying.
Visual constructions, worked problems and proofs. Guided work leads to an independent attempt, with a diagram, explanation, trace, calculation or tested artifact. Question bodies are generated when opened in your private learning path. Programming submissions are stored as text; the platform does not execute submitted code.
Use original or appropriately licensed datasets. Practical science activities use models or supplied data unless a qualified supervisor has reviewed a physical experiment.
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