Difference between revisions of "12-240/Classnotes for Tuesday September 18"

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(Recap:)
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== Various properties of fields ==
 
== Various properties of fields ==
Thrm: In a field F:
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'''Thrm 1''': In a field F:
 
1. a+b = c+b ⇒ a=c
 
1. a+b = c+b ⇒ a=c
  
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Hint: Use distributive law
 
Hint: Use distributive law
  
Thrm 2: Given a field F, there exists a map Ɩ: Z → F with the properties:  
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'''Thrm 2''': Given a field F, there exists a map Ɩ: Z → F with the properties (∀ m,n ∈ Z):  
  
 
1) Ɩ(0) =0, Ɩ(1)=1
 
1) Ɩ(0) =0, Ɩ(1)=1
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Furthermore, Ɩ is unique.
 
Furthermore, Ɩ is unique.
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'''Rough proof''':
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Test somes cases:
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Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2
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Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3
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.
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.
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.
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Ɩ(n) = 1 + ... + 1 (n times)
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Ɩ(-3) = ?
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Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)
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''What about uniqueness?'' Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.
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At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]

Revision as of 22:11, 18 September 2012

Various properties of fields

Thrm 1: 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

12. a∙b=0 iff a=0 or b=0

. . .

16. (a+b)∙(a-b)= a^2 - b^2 [Define a^2 = a∙a] Hint: Use distributive law

Thrm 2: Given a field F, there exists a map Ɩ: Z → F with the properties (∀ m,n ∈ Z):

1) Ɩ(0) =0, Ɩ(1)=1

2) Ɩ(m+n) = Ɩ(m) +Ɩ(n)

3) Ɩ(mn) = Ɩ(m)∙Ɩ(n)

Furthermore, Ɩ is unique.

Rough proof:

Test somes cases:

Ɩ(2) = Ɩ(1+1) = Ɩ(1) + Ɩ(1) = 1 + 1 ≠ 2

Ɩ(3) = Ɩ(2 +1)= Ɩ(2) + Ɩ(1) = 1+ 1+ 1 ≠ 3

. . .

Ɩ(n) = 1 + ... + 1 (n times)

Ɩ(-3) = ?

Ɩ(-3 + 3) = Ɩ(-3) + Ɩ(3) ⇒ Ɩ(-3) = -Ɩ(3) = -(1+1+1)

What about uniqueness? Simply put, we had not choice in the definition of Ɩ. All followed from the given properties.

At this point, we will be lazy and simply denote Ɩ(3) = 3_f [3 with subscript f]