that induced by inclusion in A2; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from A. Inside AN, the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. In the case of the idele group, the quotient group is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a compact group; the proof of this is essentially equivalent to the finiteness of the class number. The study of the Galois cohomology of idele class groups is a central matter in class field theory. Characters of the idele class group, now usually called Hecke characters or Größencharacters, give rise to the most basic class of L-functions. Tamagawa numbers For more general G, the Tamagawa number is defined (or indirectly computed) as the measure of G(A)/G(K). Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over K, the measure involved was well-defined: while ω could be replaced by cω with c a non-zero element of K, the product formula for valuations in K is reflected by the independence from c of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory. See also Ring of adeles References External links Topological groups Algebraic number theory Algebraic groups