14-240/Classnotes for Monday September 15
From Drorbn
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Contents |
Definition of Subtraction and Division
- Subtraction: if .
- Division: if .
Basic Properties of a Field (cont'd)
8. , .
- Proof of 8
- By F3 ,
- By F5 , ;
- By F3 , ;
- By Thm P1, .
9. s.t. ;
- s.t. .
- Proof of 9
- By F3 , .
10. .
11. .
12. or .
- Proof of 12
- <= :
- By P8 , if , then ;
- By P8 , if , then .
- => : Assume , if a = 0 we are done;
- Otherwise , by P8 , and we have ;
- by cancellation (P2) , .
.
- Proof
- By F5 ,
Theorem
s.t.
- 1. ;
- 2. ;
- 3. .
- Examples
......
In F2: