14-240/Classnotes for Monday September 15
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Definition of Subtraction and Division
- Subtraction: if [math]\displaystyle{ a, b \in F, a - b = a + (-b) }[/math].
- Division: if [math]\displaystyle{ a, b \in F, a / b = a \times b^{-1} }[/math].
Basic Properties of a Field (cont'd)
8. [math]\displaystyle{ \forall a \in F }[/math], [math]\displaystyle{ a \times 0 = 0 }[/math].
- Proof of 8
- By F3 , [math]\displaystyle{ a \times 0 = a \times (0 + 0) }[/math]
- By F5 , [math]\displaystyle{ a \times (0 + 0) = a \times 0 + a \times 0 }[/math];
- By F3 , [math]\displaystyle{ a \times 0 = 0 + a \times 0 }[/math];
- By Thm P1, [math]\displaystyle{ 0 = a \times 0 }[/math].
9. [math]\displaystyle{ \nexists b \in F }[/math] s.t. [math]\displaystyle{ 0 \times b = 1 }[/math];
- [math]\displaystyle{ \forall b \in F }[/math] s.t. [math]\displaystyle{ 0 \times b \neq 1 }[/math].
- Proof of 9
- By F3 , [math]\displaystyle{ \times b = 0 \neq 1 }[/math].
10. [math]\displaystyle{ (-a) \times b = a \times (-b) = -(a \times b) }[/math].
11. [math]\displaystyle{ (-a) \times (-b) = a \times b }[/math].
12. [math]\displaystyle{ a \times b = 0 \iff a = 0 }[/math] or [math]\displaystyle{ b = 0 }[/math].
- Proof of 12
- <= :
- By P8 , if [math]\displaystyle{ a = 0 }[/math] , then [math]\displaystyle{ a \times b = 0 \times b = 0 }[/math];
- By P8 , if [math]\displaystyle{ b = 0 }[/math] , then [math]\displaystyle{ a \times b = a \times 0 = 0 }[/math].
- => : Assume [math]\displaystyle{ a \times b = 0 }[/math] , if a = 0 we are done;
- Otherwise , by P8 , [math]\displaystyle{ a \neq 0 }[/math] and we have [math]\displaystyle{ a \times b = 0 = a \times 0 }[/math];
- by cancellation (P2) , [math]\displaystyle{ b = 0 }[/math].
[math]\displaystyle{ (a + b) \times (a - b) = a^2 - b^2 }[/math].
- Proof
- By F5 , [math]\displaystyle{ (a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b)) }[/math]
- [math]\displaystyle{ = a^2 - b^2 }[/math]
Theorem
[math]\displaystyle{ \exists! \iota : \Z \rightarrow F }[/math] s.t.
- 1. [math]\displaystyle{ \iota(0) = 0 , \iota(1) = 1 }[/math];
- 2. [math]\displaystyle{ \forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n) }[/math];
- 3. [math]\displaystyle{ \forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n) }[/math].
- Examples
[math]\displaystyle{ \iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1; }[/math] [math]\displaystyle{ \iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1; }[/math]
......
In F2: [math]\displaystyle{ \begin{align} 27 ----\gt \iota(27) &= \iota(26 + 1)\\ &= \iota(26) + \iota(1)\\ &= \iota(26) + 1\\ &= \iota(13 \times 2) + 1\\ &= \iota(2) \times \iota(13) + 1\\ &= (1 + 1) \times \iota(13) + 1\\ &= 0 \times \iota(13) + 1\\ &= 1 \end{align} }[/math]
Scanned Lecture Notes by AM
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