Last modified date: <%+ tp.file.last_modified_date() %>
- Tags:
- Refs:
- Links:
global field
function fields
- Function field: an extension $F_{/ {k}} $ where \([F: k(x)] < \infty\) for some \(x\) transcendental over \(k\).
global fields
Global fields satisfy a product formula: \begin{align*} x\in K\setminus\left\{{0}\right\}\implies \displaystyle\prod_{v\in {\operatorname{Places}}(K)} {\left\lvert {x} \right\rvert}_v = 1 \end{align*}
# local fields 
-
Idea: can arise as the rings of germs of functions, i.e. the local rings on a scheme.
-
Arise as the completions of global fields.
-
Another defintion: a field complete wrt a topology induced by a discrete valuation with a finite residue field.
-
Classification of local fields:
- Every local field is the completion of a global field wrt an absolute value.
- An archimedean local field is either \({\mathbf{R}}\) or \({\mathbf{C}}\).
- A nonarchimedean local field is a finite extension \(L/K\) for \(K={ {\mathbf{Q}}_p }\) or \({ \mathbf{F} }_q{\left(\left( t \right)\right) }\)
-
The completion of \(\operatorname{ff}(K)\) with respect to an absolute valueor valuation for \(K\) a global field is a locally compact field, and thus a local field.
-
Examples
-
Global fields
- \({\mathbf{Q}}\)
- Algebraic number fields $K_{/ {{\mathbf{Q}}}} $
- $L_{/ {K}} $ finite extensions of $K = { \mathbf{F} }_q { \left( {t} \right) } $, i.e. function fields of an algebraic curve over a finite field
-
Local fields:
- \({\mathbf{R}}\) and \({ {\mathbf{Q}}_p }\) for \(p\) all primes in \({\mathbf{Z}}\) are local.
- \({ \mathbf{F} }_q{\left(\left( t \right)\right) }\) formal Laurent series over a finite field.
- The completion of a global field at a valuation / absolute value.
- Nonexample: \({\mathbf{C}}{\left(\left( t \right)\right) }\), since its residue field is \({\mathbf{C}}{\left[\left[ t \right]\right] }/\left\langle{t}\right\rangle \cong {\mathbf{C}}\) which is not finite.
