MATH2581

**Lecturer :** Dirk Schuetz

**Term :** Michaelmas 2013

- Tuesday 4:00pm in D110
- Thursday 10:00am in W103

- Thursday 3:00pm in CG93 (even weeks only)

**Literature**

- RBJT Allenby, Rings, fields and groups : an introduction to abstract algebra, Arnold, 1991.
- PJ Cameron, Introduction to Algebra, Oxford University Press, 1998.

- MA Armstrong, Groups and Symmetry. Springer, 1988.

**Handouts**

Problem Class | Date | Solutions |

17.10. | ||

31.10. | ||

14.11. | ||

28.11. | ||

12.12. | ||

**Assignments**

Homework | Date | Hand in | Solutions |

Homework Set 1 pdf | 10.10. | 17.10. | |

Homework Set 2 pdf | 17.10. | 24.10. | |

Homework Set 3 pdf | 24.10. | 07.11. | |

Homework Set 4 pdf | 07.11. | 14.11. | |

Homework Set 5 pdf | 14.11. | 21.11. | |

Homework Set 6 pdf | 21.11. | 28.11. | |

Homework Set 7 pdf | 28.11. | 05.12. | |

Homework Set 8 pdf | 05.12. | 12.12. |

**Lecture Outline**

Date | Outline |

10.12. | In this lecture we define groups and subgroups, and see some basic examples. |

05.12. | In this lecture we study maximal ideals and use them to construct fields out of irreducible polynomials. |

28.11. | In this lecture we prove the Chinese Remainder Theorem, see some applications of it, and define prime ideals. |

26.11. | In this lecture we will see some examples of quotient rings and prove the first Isomorphism Theorem. |

21.11. | In this lecture we begin our study of quotient rings. |

19.11. | In this lecture we show Fermat's Little Theorem and analyze which prime numbers can be written as the sum of two squares. |

14.11. | In this lecture we show that Euclidean Domains are PIDs, show that the Gaussian Integers form a Euclidean Domain, and look at Unique Factorization Domains. |

12.11. | In this lecture we define an ideal, see some examples, and introduce the notion of a Principal Ideal Domain. |

07.11. | In this lecture we use the Gauss Lemma to give criteria for irreducibility of polynomials over the rationals, including the Eisenstein criterion. |

05.11. | In this lecture we show that polynomials over a field factorise uniquely into irreducibles, and consider irreducible polynomials over the complex, real and rational numbers. |

29.10. | In this lecture we deal with irreducible and prime elements, in particular for polynomials. |

24.10. | In this lecture we explain the Euclidean algorithm for polynomials, show some examples and look at roots of polynomials. |

22.10. | In this lecture we start with a more systematic analysis of the ring of polynomials over a field. |

17.10. | In this lecture we prove Bezout's identity, define fields and deal with divisors. |

15.10. | In this lecture we define kernel and image of ring homomorphisms, integral domains and units. |

10.10. | In this lecture we look at subrings, the division algorithm and ring homomorphisms. |

08.10. | In this lecture we define rings, see some examples and basic properties. |

Last modified: 12.12.2013.