How to prove $(-1)\cdot x = -x$ in a field?

Question

I am having some hard time understanding the concept of field.

I understand the 6 axioms:

1. Associativity
2. Distributivity
3. Computability
4. Closure
5. Inverses
6. Identities

The exercise I am trying to solve is the following:

Given F is a field, prove for any x belongs to F, x multiply -1 = -x.

P.S. Can anyone give me some tips regarding how to prove this type of questions that involves proofs in a field?

Answer 1

In these proofs, you should manipulate the given expression to prove in many ways until you get an idea on how to prove it.

In this question, for example, we have \begin{align*} x \cdot (-1) &= -x, \\ x \cdot (-1) + x &= 0, \\ x \cdot (-1) + x \cdot (1) &= 0. \\ \end{align*}

This last expression should give you the idea on how to solve this:

$$ x \cdot (-1) + x = x \cdot (-1) + x \cdot (1) = x \cdot (-1+1) = x \cdot 0 = 0. $$ Hence, $$ x \cdot (-1) + x = 0 \implies x \cdot (-1) = -x. $$

Answer 2

You have already been given perfectly satisfactory answers. This one is methodological.

$-1$ and $-x$ are defined in terms of addition alone. You are asked to prove that they satisfy a property related to multiplication. Now the only axioms that involve both addition and multiplication are the distributive properties, so these are the ones you should use.

Since $-1$ is defined by $$ 1 + (-1) = 0, $$ a look at the identity you have to prove suggests to multiply this by $x$, and distribute.