Office hours: M 2:30-3:30, W 12:30-2:30, or by appointment in MATH 303.

Syllabus for course

Text: K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory (Springer, 2nd edition)

Date | Section and Topic | Read before class | Homework |

August 26 | Organization and introduction | ||

August 28 | Unique Factorization in the integers | Chapter 1; Sections 1, 2 | HW 1 Due on September 9 |

August 30 | Unique factorization in k[x] | Chapter 1; Sections 2, 3 | Chpt 1: 7, 9, 18, 21, 23, 25, 26, 35-38. |

September 4 | Unique Factorization in PIDs, Euclidean Domains | Chapter 1; Sections 3, 4 | |

September 6 | Gaussian and Eisenstein Integers | Chpt1; Sect 4. Chpt 2, Sect 1 | |

September 9 | Infinitude of Primes | Chpt 2; Sections 1, 3 | HW 2 due September 18 |

September 11 | Arithmetic Functions | Chpt 2, Section 2 | Chpt 2; 1, 2, 6, 7, 8, 9, 10, 12, 15, 21, 26, 27 |

September 13 | Growth of π(x) | Chpt 2; Section 4 | |

September 16 | Congruences | Chpt 3; Sections 1, 2 | |

September 18 | Chinese Remainder Theorem | Chpt 3; Section 3, 4 | HW 3 due Sept 30: Chpt 3; 1, 3, 5, 16, 18-21; Chpt 4: 1, 6, 8, 11, 12 |

September 20 | Primitive Roots | Chpt 4; Section 1 | |

September 23 | nth-power residues | Chpt 4; Section 2 | |

September 25 | Quadratic residues | Chpt 5; Section 1 | |

September 27 | Quadratic reciprocity | Chpt 5; Section 2 | |

September 30 | Proof of quadratic reciprocity | Chpt 5; Section 3 | |

October 2 | Algebraic numbers and algebraic integers | Chpt 6, Section 1 | HW 4 due Oct 14; Chpt 5: 1, 3, 16; Chpt 6: 2, 8, 11, 13. |

October 4 | The quadratic character of 2 | Chpt 6, Section 2 | |

October 7 | The quadratic Gauss sum | Chpt 6, Section 3 | |

October 9 | Sign of the quadratic Gauss sum, a la Schur | Handout | |

October 11 | Sign of the quadratic Gauss sum, a la Kronecker | Chpt 6, Sect 4 | |

October 14 | Introduction to Finite Fields | Chpt 7, Sect 1 | |

October 16 | More on finite fields | Chpt 7, Sect 2 | Take home midterm due Oct 28 |

October 18 | Quadratic reciprocity via finite fields | Chpt 7, Sect 3 | |

October 21 | Multiplicative characters | Chpt 8, Sect 1 | |

October 23 | Gauss Sums | Chpt 8, Sect 2 | |

October 25 | Jacobi Sums | Chpt 8, Sect 3 | |

October 28 | More on Jacobi sums, Fermat equation | Chpt 8, Sect 4, 5 | |

October 30 | More on Jacobi sums | Chpt 8, Sect 6 | |

Nov 1 | Points on diagonal hypersurfaces | Chpt 8, Sect 7 | HW5 due Nov 11; Chpt 9, 19-25 |

Nov 4 | Arithmetic of Eisenstein integers | Chpt 9, Sect 1, 2 | |

Nov 6 | Cubic residues | Chpt 9, Sect 3 | |

Nov 8 | Proof of cubic reciprocity | Chpt 9, Sect 4 | |

Nov 11 | Intro to p-adic integers | B&S: 1.3.1, 1.3.2 | |

Nov 13 | p-adic integers and p-adic numbers | 1.3.2, 1.3.3 | |

Nov 15 | Convergence of p-adic numbers | 1.3.4 | |

Nov 18 | The p-adics as a metric field | 1.4 | |

Nov 20 | Congruences in the p-adics | 1.5.1 | HW 6 due Dec 4. |

Nov 22 | Hensel's Lemma | 1.5.2 | HW 6 due Dec 6: B&S p32, 2, 3, 4, 11; p40 1, 6; p 46, 1, 3 (fix the typo) |