# 14-240/Classnotes for Monday September 15

## 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}