Editing Algebraic number theory (section)

===Places===
Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on [[valuation (algebra)|valuation]]s. Consider, for example, the integers. In addition to the usual [[absolute value]] function |·| : '''Q''' → '''R''', there are [[p-adic absolute value]] functions |·|<sub>''p''</sub> : '''Q''' → '''R''', defined for each prime number ''p'', which measure divisibility by ''p''. [[Ostrowski's theorem]] states that these are all possible absolute value functions on '''Q''' (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of '''Q''' and the prime numbers.

A '''place''' of an algebraic number field is an [[equivalence class]] of [[absolute value (algebra)|absolute value]] functions on ''K''. There are two types of places. There is a <math>\mathfrak{p}</math>-adic absolute value for each prime ideal <math>\mathfrak{p}</math> of ''O'', and, like the ''p''-adic absolute values, it measures divisibility. These are called '''finite places'''. The other type of place is specified using a real or complex embedding of ''K'' and the standard absolute value function on '''R''' or '''C'''. These are '''infinite places'''. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are {{math|''r''<sub>1</sub>}} real places and {{math|''r''<sub>2</sub>}} complex places. Because places encompass the primes, places are sometimes referred to as '''primes'''. When this is done, finite places are called '''finite primes''' and infinite places are called '''infinite primes'''. If {{math|''v''}} is a valuation corresponding to an absolute value, then one frequently writes <math>v \mid \infty</math> to mean that {{math|''v''}} is an infinite place and <math>v \nmid \infty</math> to mean that it is a finite place.

Considering all the places of the field together produces the [[adele ring]] of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in the [[Artin reciprocity law]].

==== Places at infinity geometrically ====
There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let <math>k = \mathbb{F}_q</math> and <math>X/k</math> be a [[Smooth scheme|smooth]], [[Projective curve|projective]], [[algebraic curve]]. The [[Function field of an algebraic variety|function field]] <math>F = k(X)</math> has many absolute values, or places, and each corresponds to a point on the curve. If <math>X</math> is the projective completion of an affine curve <math>\hat{X} \subset \mathbb{A}^n</math> then the points in <math>X - \hat{X}</math> correspond to the places at infinity. Then, the completion of <math>F</math> at one of these points gives an analogue of the <math>p</math>-adics.

For example, if <math>X = \mathbb{P}^1</math> then its function field is isomorphic to <math>k(t)</math> where <math>t</math> is an indeterminant and the field <math>F</math> is the field of fractions of polynomials in <math>t</math>. Then, a place <math>v_p</math> at a point <math>p \in X</math> measures the order of vanishing or the order of a pole of a fraction of polynomials <math>p(x)/q(x)</math> at the point <math>p</math>. For example, if <math>p = [2:1]</math>, so on the affine chart <math>x_1 \neq 0</math> this corresponds to the point <math>2 \in \mathbb{A}^1</math>, the valuation <math>v_2</math> measures the [[order of vanishing]] of <math>p(x)</math> minus the order of vanishing of <math>q(x)</math> at <math>2</math>. The function field of the completion at the place <math>v_2</math> is then <math>k((t-2))</math> which is the field of [[power series]] in the variable <math>t-2</math>, so an element is of the form{{blockquote|<math>\begin{align}
&a_{-k}(t-2)^{-k} + \cdots  + a_{-1}(t-2)^{-1} + a_0 + a_1(t-2) + a_2(t-2)^2 + \cdots \\
&=\sum_{n = -k}^{\infty} a_n(t-2)^n
\end{align}</math>}}for some <math>k \in \mathbb{N}</math>. For the place at infinity, this corresponds to the function field <math>k((1/t))</math> which are power series of the form{{blockquote|<math>\sum_{n=-k}^\infty a_n(1/t)^n</math>}}