# Bachelor Seminar on Algebra (Summer 2018)

with Dr. Heer Zhao

We will study a series of topics which naturally continues those discussed in the lecture course Algebra 1 – Trace and Norm, Hilbert’s Theorem 90, Solvability of equations by radicals, infinite Galois theory, cyclotomic fields, the quadratic reciprocity law, simple cases of Fermat’s Last Theorem.

**Prerequisites:** Algebra 1 (Groups, Rings, Fields, Galois theory) and of course the basics (Linear Algebra 1 + 2, and maybe a bit of analysis).

**Time and place:** Tue, 14-16, S-U-3.03, First talk: April, 10.

**Contact:** ulrich.goertz@uni-due.de

**ECTS points:** 6 credit points for a successful seminar talk (in German or in English upon the choice of the speaker)

**Program:** pdf

**Organisational meeting:** February 15, 2:15pm, S-3.14. If you are interested in giving a talk in the seminar but cannot come to the organisational meeting, please send me an email.

## Talks

1 | The main theorem of Galois theory | U. Görtz |

2 | Symmetric functions | H. Zhao |

3 | The class equation and applications | O. Girnth |

4 | The Sylow theorems | L. Meurs |

5 | The fundamental theorem of algebra | D. Tambaro |

6 | The Galois group of a polynomial I | H. Zhao |

7 | The Galois group of a polynomial II | U. Görtz |

8 | Norm and trace, Hilbert’s Theorem 90 | H. Zhao |

9 | Cyclic extensions | I. Tselepidis |

10 | Cyclotomic fields and solvable extensions | A. Salarzai |

11 | The quadratic reciprocity law | F. Siethoff |

12 | The first case of Fermat’s Last Theorem for regular primes | N. N. |

13 | Topological groups | N. N. |

14 | Infinite Galois theory | N. N. |