Editing Algebraic number theory (section)

==== 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>}}