INTRODUCTION

By a Diophantine equation we shall mean an equation where we require the solution to lie in a given field (e.g. Q) or ring (e.g. ℤ). In this chapter we shall be concerned only with the first of these two cases, although often the equations concerned will be homogeneous, so the distinction between rational and integral solutions disappears.

As already noted in Chapter 4, §3 bis, a necessary condition that an equation have a solution in an algebraic number field k is that it have a solution at all the local completions k (including those at the archimedean places). (“solutions everywhere locally”). We have also given (Chapter 4, §3 bis; Chapter 10, §9) examples where this condition is not sufficient. There are however some general situations where the existence of solutions everywhere locally implies the existence of a global solution: when this is the case, there is said to be a Hasseprinciple.

Perhaps the two most important examples of Hasse principles are

THEOREM A. (Hasse). Let

be a quadratic form, where k is an algebraic number field. Suppose that the equation F = 0 has a nontrivial solution everywhere locally. Then it has a nontrivial solution in k.

Note The trivial solution of F = 0 is, of course, that in which all the variables are 0.

THEOREM B. (Hasse). Let K/k be a cyclic extension of number fields and let b ∈ k. If. b is a norm everywhere locally then it is a norm globally.