12-240/Classnotes for Tuesday September 18

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Recap:

Thrm: In a field F: 1. a+b = c+b => a=c

2. b≠0, a∙b=c∙b => a=c

3. 0 is unique.

4. 1 is unique.

5. -a is unique.

6. a^-1 is unique (a≠0)

7. -(-a)=a

8. (a^-1)^-1 =a

9. a∙0=0 **Surprisingly difficult, required distributivity.

10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1

11. (-a)∙(-b)=a∙b