Zero Products and Elementary Identities in a Field

Let $F$ be a field, with additive identity $0$, multiplicative identity $1$, additive inverse $-x$ of an element $x$, and multiplicative inverse $x^{-1}$ of an element $x\ne0$. Write $x-y=x+(-y)$ and $x^{2}=x\cdot x$; a sum of three terms is written without brackets, which is unambiguous by axiom 1 of that definition. Let $x,y\in F$. Then the following hold.

1. (Annihilation) $0x=x0=0$.

2. (Signs) $(-x)y=-(xy)$ and $(-x)(-y)=xy$.

3. (No zero divisors) If $xy=0$, then $x=0$ or $y=0$.

4. (Difference of squares) $(x-y)(x+y)=x^{2}-y^{2}$.

5. (Square of a sum) $(x+y)^{2}=x^{2}+(xy+xy)+y^{2}$.