Schedule

Day | Topics | Reading |
---|---|---|

Tu. 8/28 | Introduction: Groups, Rings, Fields. Groups, subgroups and homomorphisms. |
[OS] Sec. 1. [A] Sec 1.1 |

Th. 8/30 | Order theorem. Lattice of subgroups. |
[OS] Sec. 1-2 [A] Sec. 1.2 |

Tu. 9/4 | The symmetric Group. Groups in sage. Meet in lab GMCS 422 |
[JB] Ch 3-6 Sage exercises. |

Th. 9/6 | Cartesian product of groups. Automorphism groups. The alternating group |
[OS] Sec. 1-2 [A] Sec. 1.3 |

Tu. 9/11 | Cosets, index of a group,Cayley's theorem. Conjugation. |
[OS] Sec. 3 |

Th. 9/13 | Rings and unit groups. | [OS] Sec. 4-5. [H] Sec. 3.1-2 |

Tu. 9/18 | The main theorems for Polynomial Rings. | [OS] Sec. 8-10. [H] Ch 4,5. |

Th. 9/20 | Polynomial ring modulo a polynomial. | [OS] Sec. 8-10. [H] Ch 4,5. |

Tu. 9/25 | More on polynomial rings Mason Stothers theorem and the abc conjecture. |
[OS] Sec 10. |

Th. 9/27 | Irreducibility in polynomial rings.
Quotients of . F[x] |
[OS] Sec. 10-11 [H] Sec. 4.5. |

Tu. 10/2 | Fields, homomorphisms, and automorphism groups.
The quadratic formula and quadratic extensions of
Q |
[OS] Sec 12-13. |

Th. 10/4 | The cubic formula and cubic extensions of . Q |
[OS] Sec. 13-14. |

Tu. 10/9 | Finite Fields. | [OS] Sec. 15. |

Th 10/11 | Finite Field structure. Automorphisms and containment |
[OS] Sec. 15. |

Tu 10/16 | Normal subgroups and quotient groups
The factor theorem.
The 1st and 3rd isomorphism theorems and the correspondence theorem. |
[A] Sec. 1.4 |

Th 10/18 | The second isomorphism theorem. Direct and semidirect products. | [A] Sec. 1.4-5 |

Tu 10/23 | Direct and semidirect products.
Exact sequences and classification of groups. |
[A] Sec. 1.4-5 |

Th. 10/25 | Abelian groups.
Free groups. Generators and relations |
[H] 8.2 (or other sources) [A] Sec. 5.8 |

Tu. 10/30 | Problem solving party. | . |

Th. 11/1 | TEST | . |

Tu. 11/6 | Groups acting on sets Orbit/stabilizer theorem. Groups of prime-power order. |
[A] 5.1-2. |

Th. 11/8 | Proof and applications of the Sylow theorems. | [A] 5.4-5. |

Tu. 11/13 | Algebraic field extensions.
Splitting fields and algebraic closure. |
[A] 3.1-2; [H] 10.4. |

Th. 11/15 | Separable extensions. Primitive element Theorem. |
[A] 3.3-5 |

Tu. 11/20 | Normal and Galois extensions. The Galois correspondence. |
[A]3.5, 6.1-2; [H] 12.2. |

Tu. 11/27 | The fundamental theorem of Galois theory. Cyclotomic fields extensions. |
[A] 6.2; [H] 12.2. [A] 6.5 |

Th 11/29 | Cyclic and abelian extensions. | [A] 6.7. |

Tu 12/4 | Geometric constructions. | [A] 6.8. |

Th 12/6 | Commutator subgroups and solvable groups. Solvability by radicals |
[A] 5.7. |

[A] Ash,

[H] Hungerford,

[OS] O'Sullivan,

[JB] Judson, Beezer