Let $F$ be a field and $x, y\in F$. Prove:

Question

Use field axioms to prove:

a) $(−1) · (−x) = x $

b) If $x · y = 0$ then $x = 0$ or $y = 0$

I don't understand how to approach these questions. Does the field include $1$ and $0$ as well?

Answer 1

For $a)$, we know that $(-x)+x=(-1)+1=0$, so

$(-1)(-x)=(-1)(-x)+(-x)+x=(-1+1)(-x)+x=0+x=x$

For $b)$ suppose that, for instance, $x \neq 0$. Since $F$ is a field, there exists $z \in F$ such that $z \cdot x= 1$. Hence: $x \cdot y=0$ $\Rightarrow$ $z (\cdot x \cdot y)=z \cdot 0$ $\Rightarrow$ $(z \cdot x) \cdot y=0$$\Rightarrow$ $y=0$

Answer 2

a) $$(1-1)\cdot(-x)=0$$ Add $x$ to both sides: $$x+(1-1)\cdot(-x)=x$$ $x$ is the additive inverse of $(-x)$ so those two can cancel when added: $$(-1)\cdot(-x)=x$$

b) Suppose $x\cdot y=0$ and $x \neq 0$. Since this is a field, $x$ has an inverse. $$x\cdot y=0$$ $$y=x^{-1}\cdot0$$ $$y=0$$ Hence if the product of two elements is $0$ and one of them is non-zero, then the other must be zero.