14-240/Classnotes for Monday September 15

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Definition of Subtraction and Division

  • Subtraction: if a, b \in F, a - b = a + (-b).
  • Division: if a, b \in F, a / b = a \times b^{-1}.

Basic Properties of a Field (cont'd)

8. \forall a \in F, a \times 0 = 0.

Proof of 8
By F3 , a \times 0 = a \times (0 + 0)
By F5 , a \times (0 + 0) = a \times 0 + a \times 0;
By F3 , a \times 0 = 0 + a \times 0;
By Thm P1, 0 = a \times 0.

9. \nexists b \in F s.t. 0 \times b = 1;

\forall b \in F s.t. 0 \times b \neq 1.
Proof of 9
By F3 , \times b = 0 \neq 1.

10. (-a) \times b = a \times (-b) = -(a \times b).

11. (-a) \times (-b) = a \times b.

12. a \times b = 0 \iff a = 0 or b = 0.

Proof of 12
<= :
By P8 , if a = 0 , then a \times b = 0 \times b = 0;
By P8 , if b = 0 , then a \times b = a \times 0 = 0.
=> : Assume a \times b = 0 , if a = 0 we are done;
Otherwise , by P8 , a \neq 0 and we have a \times b = 0 = a \times 0;
by cancellation (P2) , b = 0.

(a + b) \times (a - b) = a^2 - b^2.

Proof
By F5 , (a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))
= a^2 - b^2

Theorem

\exists! \iota : \Z \rightarrow F s.t.

1. \iota(0) = 0 , \iota(1) = 1;
2. \forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n);
3. \forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n).
Examples

\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1; \iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;

......

In F2: 
\begin{align}
27 ----> \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}

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