discrete in is a discrete subgroup of and with the same argument, is discrete in An alternative definition is: where is a subring of defined by As a consequence, contains all elements which fulfil for all Lemma 1. Let The following set is finite: Proof. Define is compact and the set described above is the intersection of with the discrete subgroup in and therefore finite. Lemma 2. Let be set of all such that for all Then the group of all roots of unity of In particular it is finite and cyclic. Proof. All roots of unity of have absolute value so For converse note that Lemma 1 with and any implies is finite. Moreover for each finite set of places Finally suppose there exists which is not a root of unity of Then for all contradicting the finiteness of Unit Theorem. is the direct product of and a group isomorphic to where if and if Dirichlet's Unit Theorem. Let be a number field. Then where is the finite cyclic group of all roots of unity of is the number of real embeddings of and is the number of conjugate pairs of complex embeddings of It stands, that Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let be a number field. We already know that set and note Then we have: Approximation theorems Weak Approximation Theorem. Let be inequivalent valuations of Let be the completion of with respect to Embed diagonally in Then is everywhere dense in In other words, for each and for each there exists such that: Strong Approximation Theorem. Let be a place of Define Then is dense in Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if we omit one place (or more), the property of discreteness of is turned into a denseness of Hasse principle Hasse-Minkowski Theorem. A quadratic form on is zero, if and only if, the quadratic form is zero in each completion Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle (also known as local–global principle) is to solve a given problem of a number field by doing so in its completions and then concluding on a solution in Characters on the adele ring Definition. Let be a locally compact