definition of field
Topics
About: field
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a definition of field
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*F\): \(\in \{\text{ the rings }\}\)
//
Conditions:
\(\forall r_1, r_2 \in F (r_1 r_2 = r_2 r_1)\)
\(\land\)
\(\forall r \in F (\exists r' \in F (r r' = r' r = 1))\)
//
2: Note
Inevitably, such \(r'\) is unique for fixed \(r\), because supposing there is another \(r'' \in F\) such that \(r r'' = r'' r = 1\), from \(r' r = 1\), \(r' r r'' = 1 r'' = r''\), but \(r' r r'' = r' 1 = r'\), and so, \(r' = r''\).
As \(r'\) is unique for \(r\), it is warranted to be denoted as \(r^{-1}\), called "the inverse of \(r\)".
