Zero Products and Elementary Identities in a Field

Let $F$ be a \reftext{def:field-c54-2026b}{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.

\textbf{1. (Annihilation)} $0x=x0=0$.

\textbf{2. (Signs)} $(-x)y=-(xy)$ and $(-x)(-y)=xy$.

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

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

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