14-240/Classnotes for Monday September 15

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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:

Scanned notes

http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2) http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)