An introduction to algebraic number theory, with emphasis on quadratic fields. In such fields the familiar unique factorisation enjoyed by the integers may fail, but the extent of the failure is measured by the class group.
The following topics will be treated with an emphasis on quadratic fields: Field extensions, minimum polynomial, algebraic numbers, conjugates and discriminants, Gaussian integers, algebraic integers, integral basis, quadratic fields, cyclotomic fields, norm of an algebraic number, existence of factorisation. Factorisation in quadratic fields. Ideals, Z-basis, maximal ideals, prime ideals, unique factorisation theorem of ideals and consequences, relationship between factorisation of numbers and of ideals, norm of an ideal. Ideal classes, finiteness of class number, computations of class number. Additional material: The topics of M3P15, treated for general number fields, together with Fractional ideals, MinkowskiÕs theorem on linear forms, Ramification, characterisation of units of cyclotomic fields, a special case of FermatÕs last theorem.