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Nydus Open Learning

Mathematics

Mathematics · Avançado · Nível 3

Abstract algebra I

Binary operations and the group axioms; Cayley tables; what the axioms already force; the order of an element, subgroups and the subgroup test, and the classification of cyclic groups; permutations in cycle notation, composition and inverses, order and parity, and the alternating group; cosets, Lagrange's theorem with its corollaries and normal subgroups; homomorphisms, kernels and images, quotient groups and the first isomorphism theorem; rings, integral domains and fields, units and zero divisors, ideals and quotient rings, prime and maximal ideals; and polynomial rings over a field — division, roots, irreducibility and the construction of the finite fields.

Ages 18+ 26 lessons about 12 hours in all

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As lições

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  1. 1 Groups and the group axioms
  2. 2 Cayley tables
  3. 3 What the axioms already force
  4. 4 The order of an element
  5. 5 Subgroups and the subgroup test
  6. 6 Cyclic groups
  7. 7 Permutations and cycle notation
  8. 8 Composing permutations and inverses
  9. 9 Order and parity of a permutation
  10. 10 The alternating group and Cayley's theorem
  11. 11 Cosets
  12. 12 Lagrange's theorem
  13. 13 The corollaries of Lagrange
  14. 14 Normal subgroups
  15. 15 Homomorphisms and isomorphisms
  16. 16 Kernels and images
  17. 17 Quotient groups
  18. 18 The first isomorphism theorem
  19. 19 Rings, integral domains and fields
  20. 20 Units and zero divisors
  21. 21 Ideals and quotient rings
  22. 22 Prime and maximal ideals
  23. 23 Polynomial rings and division
  24. 24 Roots and the factor theorem
  25. 25 Irreducibility
  26. 26 Finite fields