Prove that $-1 \cdot x=-x$

Question

While working on a proof for class, I came to a point where I couldn't go any further without knowing that $-1 \cdot x=-x$. Is there a way to prove this using the axioms of a field?

Answer 1

Of course.

First prove that $0\cdot x = 0$ using the distributive property.

$0\cdot x = (0+0) \cdot x = 0\cdot x + 0\cdot x \Longrightarrow 0\cdot x = 0$

Then try to prove $(-1)\cdot x = -x$ using the distributive property again, but in a different form.

$0 = 0\cdot x = (1+(-1))\cdot x = 1\cdot x + (-1)\cdot x = x + (-1)\cdot x \Longrightarrow (-1)\cdot x = -x$

Answer 2

Assume we already know that $0\cdot x=0$. Then

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

But that means, that $(-1)\cdot x$ is the additive inverse of $x$, i.e.

$$(-1)\cdot x=-x$$