Why is $0\cdot x$ defined in fields even though $F\setminus\{0\}$ is the set for multiplication

Question

I am somewhat confused about a definition thing about fields. It says that $\otimes$ is defined on $F \setminus \{0\}$, but in all tables I find, the zero row and column do have entries. From $\otimes$ beig defined for $F \setminus \{0\}$ I would expect the $0$ element to not occur in multiplication tables. Consider these tables: https://en.wikipedia.org/wiki/Finite_field#Field_with_four_elements

Why is $0$ defined in the multiplication table or conversely if $0$ is defined for multiplication why does it say in the definition that $\otimes$ is defined on $F \setminus \{0\}$?

Answer 1

You are wrong, multiplication is defined on whole $F$. You are confusing it with existence of inverses for multiplication. They exist on $F\setminus\{0\}$, i.e. all elements of $F$ are invertible with respect to multiplication except $0$. This does not mean that multiplication with $0$ is not defined, it means that division with $0$ is not defined! One not only has mulitplication with $0$, from distributivity you have: $$0\cdot x = (0+0)\cdot x = 0\cdot x + 0\cdot x\implies 0\cdot x = 0$$ for all $x\in F$.

Answer 2

Multiplication is defined on all of $F$. What isn't true is that $F$ with multiplication is a group (however $F\setminus \{0\}$ is).