Algebraic number theory

Description

Resultants......Page 7 Algebraic elements, algebraic extensions......Page 9 The degree of an extension......Page 13 The trace, norm, and characteristic polynomial......Page 15 Primitive elements......Page 17 Extension of complex embeddings......Page 19 Traces and norms vs. complex embeddings......Page 20 Monic polynomials......Page 23 The ring of integers......Page 24 Linear algebra over Z......Page 25 Orders......Page 28 Discriminants, part I......Page 29 Denominators vs. the index......Page 33 Discriminants, part II......Page 34 The case of quadratic fields......Page 38 The case of cyclotomic fields......Page 39 Ideals and factorisation......Page 41 Reminder on ideals......Page 42 Integral closure......Page 44 Dedekind domains......Page 46 Factorisation theory in Dedekind domains......Page 48 Decomposition of primes......Page 51 Practical factorisation......Page 53 Ramification......Page 58 The case of quadratic fields......Page 61 The case of cyclotomic fields......Page 63 UFDs. vs. PID. vs. Dedekind domains......Page 66 Ideal inversion......Page 67 The class group......Page 68 Finiteness of the class group: the Minkowski bound......Page 70 Sums of two squares......Page 72 Another norm equation......Page 73 A norm equation with a nontrivial class group......Page 74 Mordell equations......Page 76 The regular case of Fermat's last theorem......Page 78 Units in a domain......Page 79 Units in ZK......Page 80 Roots of unity under complex embeddings......Page 82 Bounding the size of WK......Page 84 Dirichlet's theorem......Page 85 The case of quadratic fields......Page 87 The case of cyclotomic fields......Page 89 The Pell–Fermat equation......Page 90 Class groups of real quadratic fields......Page 91 Lattices......Page 94 Minkowski's theorem......Page 98 Applications to number theory......Page 99 Discriminant and ring of integers......Page 104 Class group and units......Page 105 Complete examples......Page 108