Tell whether or not $\displaystyle \mathbb{Q}$ with the two operations ⊕ and $\displaystyle \cdot$ (in fact I want to type the $\displaystyle \cdot$ but with a circle as in ⊕ but I don't know how to) is a field.

Looking to my class notes, a field $\displaystyle \mathbb{K}$ is a set with 2 operations "$\displaystyle +$" and "$\displaystyle \cdot$" such that

1)$\displaystyle x+y=y+x$

2)$\displaystyle (x+y)+z=x+(y+z)$

3)$\displaystyle \exists{!}$ element "$\displaystyle 0$" such that $\displaystyle x+"0"=x$

4)$\displaystyle \forall x\in \mathbb{K}$, $\displaystyle \exists{!} "-x"$ such that $\displaystyle x+(-x)=0$

5)$\displaystyle x\cdot y=y\cdot x$

6)$\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z)$

7)$\displaystyle \exists{!}$ element $\displaystyle 1$ such that $\displaystyle x\cdot 1=x$

8)$\displaystyle \forall x\neq 0$, $\displaystyle \exists{!} x^{-1}$ such that $\displaystyle x\cdot x^{-1}=1$

9)$\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z$.

So to solve my problem, I have to check if any $\displaystyle x$ and $\displaystyle y$ elements of $\displaystyle \mathbb{Q}$ respect the 9 rules. But I'm not able to start. How can I check if any element $\displaystyle x$ and $\displaystyle y$ of $\displaystyle \mathbb{Q}$ are such that $\displaystyle x+y=y+x$? I thought about writing down $\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{b+a}{a\cdot}$ and $\displaystyle \frac{1}{b}+\frac{1}{a}=...$ But it doesn't solve the problem. (Note that $\displaystyle a$ and $\displaystyle b \in \mathbb{Z}$).

I also see that a field is a set with 3 elements as minimum. (unless $\displaystyle x$,$\displaystyle y$ an $\displaystyle z$ can be the same element.)