House 2, Room 138,

Tel. 018 4713284,

The following sections will be covered in the course (sections in in parenthesis will be covered if time permits):

1.1 - 1.5, 2.1 - 2.4, 3.1 - 3.3, 5.1 - 5.6, 6.1 - 6.2, 7.1 - 7.2, 8.1, 8.3, 8.5, 9.1 - 9.4, 9.6

More precise plan for the first lectures:

2. 9. | Binary Operations (1.1), isomorphic binary structures (1.2) |

3. 9. | Groups (1.3), subgroups (1.4) |

6. 9. | Cyclic groups and generators (1.5) |

7. 9. | Groups of permutations (2.1.), Orbits, cycles and the alternating groups (2.2) |

9. 9. | Cosets and the theorem of Lagrange (2.3) |

10. 9. | Direct products and finitely generated abelian groups (2.4) |

13. 9. | Homomorphisms (3.1), factor groups (3.2) |

16. 9. | Factor group computations and simple groups (3.3) |

17. 9. | Series of groups (3.4), rings and fields (5.1) |

20. 9. | Integral domains (5.2), Fermat's theorem, Euler's theorem (5.3) |

21. 9. | Quotient fields (5.4), polynomial rings (5.5) |

23. 9. | Factorizations of polynomials over a field (5.6) |

24. 9. | Homomorphisms and factor rings (6.1), Prime and maximal ideals (6.2) |

1. 10 | Unique factorization domains (7.1), Eucledian domains (7.2) |

4. 10. | Introduction to extension fields (8.1), Algebraic Extensions (8.3) |

5. 10. | Finite fields (8.5), automorphisms of fields (9.1) |

7. 10. | The isomorphism extension theorem (9.2) |

11. 10. | Splitting fields (9.3), Separable extensions (9.4) |

12. 10. | Galois theory (9.6) |

15. 10. | Illustrations of Galois theory (9.7) |

Some slightly more difficult but very interesting examples (examples to be discussed in the example classes are in

- Section 1.1: 37
- Section 1.2: 25, 26, 27
- Section 1.3: 27, 28, 30, 33, 37, 38
- Section 1.4:
**49**, 53, 54,**57** - Section 1.5: 55,
**56**,**62** - Section 2.1:
**38 - 41**(with proofs; for which A,B do you get a group, for which not?);**45**, 46 - Section 2.2:
**24 a, b (but n at least 3)**,**26**,**31** - Section 2.3:
**28**,**35**, 45, 46 (45, 46 are a little bit longer) - Section 2.4:
**48**, 52 - Section 3.1:
**46**, 49, 50,**53** - Section 3.2: 24,
**31** - Section 3.3: 37 (relatively long),
**38, 39, 40** - Section 3.4: 20,
**22** - Section 5.1: 32, 37, 42,
**46** - Section 5.2:
**21**, 22,**26**, 28 - Section 5.3: 25,
**26** - Section 5.4: 12, 13
- Section 5.5: 24, 30,
**31 c** - Section 5.6:
**32**, 34 - Section 6.1: 17,
**20**,**28**, 34 - Section 6.2: 24,
**28**, 30,**31**, 32, 34 - Section 7.1: 25, 26,
**30**,**32**, 33, 34 - Section 7.2:
**7**,**18** - Section 8.1:
**28**, 32,**33** - Section 8.3: 21,
**24**,**26**, 29,**34** - Section 8.5:
**12**,**15** - Section 9.1: 32,
**34**,**35** - Section 9.2: 9,
**12** - Section 9.3: 17,
**18**, 19,**20**,**21**, 22, 23, 24 - Section 9.4: 9, 12,
**14** - Section 9.6: 17, 18,
**19**, 21, 22,**26**