Algebraic Number Theory - Basic Notions - Primes and Places

Primes and Places

An important generalization of the notion of prime ideal in O is obtained by passing from the so-called ideal-theoretic approach to the so-called valuation-theoretic approach. The relation between the two approaches arises as follows. In addition to the usual absolute value function |·| : Q → R, there are absolute value functions |·|_p : Q → R defined for each prime number p in Z, called p-adic absolute values. Ostrowski's theorem states that these are all possible absolute value functions on Q (up to equivalence). This suggests that the usual absolute value could be considered as another prime. More generally, a prime of an algebraic number field K (also called a place) is an equivalence class of absolute values on K. The primes in K are of two sorts: -adic absolute values like |·|_p, one for each prime ideal of O, and absolute values like |·| obtained by considering K as a subset of the complex numbers in various possible ways and using the absolute value |·| : C → R. A prime of the first kind is called a finite prime (or finite place) and one of the second kind is called an infinite prime (or infinite place). Thus, the set of primes of Q is generally denoted { 2, 3, 5, 7, ..., ∞ }, and the usual absolute value on Q is often denoted |·|_∞ in this context.

The set of infinite primes of K can be described explicitly in terms of the embeddings K → C (i.e. the non-zero ring homomorphisms from K to C). Specifically, the set of embeddings can be split up into two disjoint subsets, those whose image is contained in R, and the rest. To each embedding σ : K → R, there corresponds a unique prime of K coming from the absolute value obtained by composing σ with the usual absolute value on R; a prime arising in this fashion is called a real prime (or real place). To an embedding τ : K → C whose image is not contained in R, one can construct a distinct embedding τ, called the conjugate embedding, by composing τ with the complex conjugation map C → C. Given such a pair of embeddings τ and τ, there corresponds a unique prime of K again obtained by composing τ with the usual absolute value (composing τ instead gives the same absolute value function since |z| = |z| for any complex number z, where z denotes the complex conjugate of z). Such a prime is called a complex prime (or complex place). The description of the set of infinite primes is then as follows: each infinite prime corresponds either to a unique embedding σ : K → R, or a pair of conjugate embeddings τ, τ : K → C. The number of real (respectively, complex) primes is often denoted r₁ (respectively, r₂). Then, the total number of embeddings K → C is r₁+2r₂ (which, in fact, equals the degree of the extension K/Q).