How to show that $(-a)(-b)=ab$ holds in a field?

Question

Let $\mathcal{F}$ be a field. How do I prove that $$\forall a,b \in \mathcal{F}:(-a)(-b)=ab.$$

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velut luna · Answer 1 · 2018-11-15T13:42:51.253

5

$$(a+(-a))b=0$$ $$ab+(-a)b=0$$ $$(-a)b=-(ab)$$

Now

$$(-a)(b+(-b))=0$$ $$(-a)b+(-a)(-b)=0$$ $$-(ab)+(-a)(-b)=0$$ $$(-a)(-b)=ab$$

edited Nov 15 '18 at 13:42

answered Aug 26 '16 at 14:50

velut luna

score 0 · Answer 2 · answered Aug 26 '16 at 14:49

0

Hint: Proof that $$(-a)(-b)+(-(ab))=0$$

answered Aug 26 '16 at 14:49

AsdrubalBeltran

2 Answers2