This work gives a full description of a method for analyzing the admissible complex representations of the general linear group G Gl N F of a non Archimedean local field F in terms of the structure of these representations when they are restricted to certain compact open subgroups of G The authors define a family of representations of these compact open subgroups which they call simple types The first example of a simple type the trivial type is the trivial character of an Iwahori subgroup of G The irreducible representations of G containing the trivial simple type are classified by the simple modules over a classical affine Hecke algebra Via an isomorphism of Hecke algebras this classification is transferred to the irreducible representations of G containing a given simple type This leads to a complete classification of the irreduc ible smooth representations of G including an explicit description of the supercuspidal representations as induced representations A special feature of this work is its virtually complete reliance on algebraic methods of a ring theoretic kind A full and accessible account of these methods is given here