Algebraic Number Field - Places

Places

Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number. This situation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field F into its various topological completions at once.

A place of a number field F is an equivalence class of absolute values on F. Essentially, an absolute value is a notion to measure the size of elements f of F. Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). In general, they fall into three regimes. Firstly (and mostly irrelevant), the trivial absolute value | |₀, which takes the value 1 on all non-zero f in F. The second and third classes are Archimedean places and non-Archimedean (or ultrametric) places. The completion of F with respect to a place is given in both cases by taking Cauchy sequences in F and dividing out null sequences, that is, sequences (x_n)_{n ∈ N} such that |x_n| tends to zero when n tends to infinity. This can be shown to be a field again, the so-called completion of F at the given place.

For F = Q, the following non-trivial norms occur (Ostrowski's theorem): the (usual) absolute value, which gives rise to the complete topological field of the real numbers R. On the other hand, for any prime number p, the p-adic absolute values is defined by

|q|_p = p−n, where q = pn a/b and a and b are integers not divisible by p.

In contrast to the usual absolute value, the p-adic norm gets smaller when q is multiplied by p, leading to quite different behavior of Q_p vis-à-vis R.