description/proof of that field is integral domain
Topics
About: field
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of field.
- The reader knows a definition of integral domain.
Target Context
- The reader will have a description and a proof of the proposition that any field is an integral domain.
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:
\(F\): \(\in \{\text{ the fields }\}\)
//
Statements:
\(F \in \{\text{ the integral domains }\}\)
//
2: Natural Language Description
For any field, \(F\), \(F\) is an integral domain.
3: Proof
Whole Strategy: Step 1: see that \(F\) is a nonzero commutative ring; Step 2: see that for any 2 elements of \(F\) such that the product is \(0\), one of the elements is \(0\).
Step 1:
\(F\) is a nonzero commutative ring, by the definition of field.
Step 2:
Let us take any elements, \(r_1, r_2 \in F\). Let us suppose that \(r_1 r_2 = 0\). If \(r_2 \neq 0\), \(r_1 = r_1 r_2 {r_2}^{-1} = 0 {r_2}^{-1} = 0\); if \(r_1 \neq 0\), \(r_2 = {r_1}^{-1} r_1 r_2 = {r_1}^{-1} 0 = 0\). That means that \(r_1 = 0\) or \(r_2 = 0\).
As the logical contraposition, \(\lnot (r_1 = 0 \lor r_2 = 0) = (r_1 \neq 0 \land r_2 \neq 0) \implies r_1 r_2 \neq 0\).
