Quiz 1 | August 21, 2015 |

Quiz 2 | September 11, 2015 |

Mid Sem | September 29, 2015 |

Quiz 3 | October 16, 2015 |

Quiz 4 | November 6, 2015 |

Quiz 5 | November 18, 2015 |

End Sem | November 20, 2015 |

Ian Stewart : Galois Theory

Patrick Morandi : Field and Galois Theory

Serge Lang : Algebra

Dummit and Foote : Abstract Algebra

Mid Sem | 30 |

End Sem | 30 |

Quizzes: best 4 of 5 quizzes, 10 pts each | 40 |

Total | 100 |

Week | Date | Content |
---|---|---|

1 | August 3 | Review of ring theory; introduction to Galois theory |

August 5 | Fields, field extensions, Examples | |

August 7 (t) | Existence of finite fields of different sizes. Examples of irreducible polynomials. Some properties of prime and maximal ideals. Finite integral domain is a field. | |

2 | August 10 | A brief review of ring theory: Definitions, domains, fields, homomorphisms, ideals, quotients. Number of elements in a finite field is a prime power. |

August 12 | Isomorphism theorems. Polynomial rings over a field. gcd, lcm and their properties. | |

August 14 (t) | Computed kernels of some morphisms. The problem set is here. | |

3 | August 17 | Existence of a field extension in which a polynomial splits, prime fields, examples. |

August 19 | Clarified some doubts; existence of a finite field of size \(p^n\), some lemma's which will lead to Gauss' lemma. | |

August 21 (t) | Quiz 1. Problem set. | |

4 | August 24 | Eisenstein criterion. Solution of cubic and quartic polynomials. |

August 26 | Extensions, splitting fields, adjoining roots. Degree formula | |

August 28 (t) | Problem set. | |

5 | August 31 | Lifting isomorphisms to splitting fields. |

September 2 | Separability. Number of extensions. Galois group. | |

September 4(t) | Problem set. | |

6 | September 7 | Galois groups of intermediate extensions. Finite subgroups of the multiplicative group are cycle. Irreduciblility in \(\mathbb{Q}[X]\) and irreducibility mod \(p\) |

September 9 | Computed \(\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\) and Galois group of cyclotomic extenstions. | |

September 11(t) | Problem set. | |

7 | September 14 | Solvable groups and radical extensions |

September 16 | Polynomial being solvable by radicals implies that the Galois group of the splitting field is solvable. | |

September 18(t) | Problem set. Quiz 2 | |

8 | September 28 | Mid Sem |

September 30 | Class postponed because of math symposium. | |

October 2(t) | Holiday: Gandhi Jayanti | |

9 | October 5 | Linear independence of characters |

October 7 | Criteria for an extension to be Galois. | |

October 9(t) | An extra class on some lemmas leading to fundamental theorem of Galois theory, and a tutorial on symmetric functions. Problem set. | |

10 | October 12 | Fundamental theorem of Galois theory. Simple extensions and finiteness of number of intermediate fields. |

October 14 | Some corollaries to the Fundamental theorem. Fundamental theorem of algebra. | |

October 16(t) | Problem set. Quiz 3. | |

11 | October 19 | Norms. Norms as determinant of the multiplication map. |

October 21 | Hilbert's theorem 90 and some corollaries. | |

October 23 | Festival break. | |

12 | October 26 | A polynomial is solvable using radicals iff the Galois group is solvable. |

October 28 | Discriminants and computing Galois groups of splitting fields of low degree polynomials over \(\mathbb{Q}\). | |

October 31 (t) | A set of exercises leading to computing the Galois group of the Artin Schreier polynomial. | |

13 | November 2 | Finding Galois group of quartics over \(\mathbb{Q}\). Resolvent cubic. |

November 4 | Normality and separability in infinite extensions. Infinite Galois extensions. | |

November 6 (t) | Quiz 4. | |

14 | November 9 | Some comments on covering space theory, Galois theory and theory of Riemann surfaces. |

November 11 | Holiday: Diwali | |

November 13 | Ruler and compass constructions: impossibility of trisection of an angle, doubling a cube and squaring a circle. | |

15 | November 16 (t) | Some problems regarding computing Galois groups. |

November 18 | Quiz 5. | |

November 20 | End Sem. |