MATH 242: Algebraic number theory

1 A review of some algebra......Page 2
2 Quadratic residues and quadratic reciprocity......Page 4
3 Algebraic numbers and algebraic integers......Page 12
4 Algebraic number fields......Page 18
4.1 First example: Gaussian integers......Page 19
4.3 Third example: Z[-5]......Page 20
4.4 Introductory tools for studying number fields: Norm, Trace, Discriminant, and Integral bases......Page 21
4.4.1 Norm and Trace......Page 22
4.4.2 Discriminant......Page 24
4.4.3 Integral bases......Page 25
4.4.4 Applications of integral bases to number fields......Page 26
5 Main theoretic properties of DF......Page 27
5.1 The class group, its finiteness, and cancellation of ideals......Page 28
5.2 Dedekind domains and Unique factorisation of ideals......Page 31
5.3 Norms of ideals......Page 34
6 Explicitly constructing ideals in DF and generators of ClF......Page 37
7 Calculations of class groups of quadratic extensions, and applications......Page 42
7.1 d=-5......Page 43
7.4 d=-10......Page 46
7.6 d=-14......Page 47
7.10 d=-30......Page 48
7.11 d=2, 3, 5, 6, 7, 11, 13, 17, 21, 29, 33, 37, 41......Page 49
8.1 Q() and its ring of integers.......Page 50
8.2 Fermat\'s Last Theorem......Page 53
9 Ramification theory......Page 56
9.1 Ramification in quadratic extensions and quadratic reciprocity......Page 58
10 A new proof of quadratic reciprocity......Page 62