Proof that there exist such an element in Q

[fluidistic]

Homework Statement

I'm asked to say whether the set Q,+,. is a field.
To be a field it must respect 8 conditions. And one of them is that there exists a unique element -x in Q such that x+(-x)=0 for all x in Q. I realize I have to prove the existence of -x but also its uniqueness. For the existence I don't know how could I approach it. For the uniqueness I'm sure that I could do it by absurd. That is by suposing that there exist more than one element -x that satisfies the same property and fall into a contradiction.
Can you get me started or help to get started for showing the existence?
Thank you!

 

[tiny-tim]

I'm asked to say whether the set Q,+,. is a field.
…
I realize I have to prove the existence of -x but also its uniqueness. For the existence I don't know how could I approach it.

Hi fluidistic!

(good sig! )

If Q is the rationals, then define the inverse of p/q as -p/q.

 

[latentcorpse]

assume there is an element -x such that x + (-x) = 0

now assume there is an element y such that x + y =0
This implies y=-x therefore we conclude that

-x is the unique element

 

[fluidistic]

Ok thank you both, I think I got it. Wasn't that hard it seems!