14-240/Classnotes for Monday September 15
From Drorbn
|
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: