Galois Field $GF(6)$

Question

The set called $GF(6)=\{0,1,2,3,4,5\}$ also has the mathematical operations addition modulo $6$ and multiplication modulo $6$. We want to prove that this is not a field, since it does not match with the law of fields. In which one do we see the mistake?

Answer 1

Hint: In this one:$$(\forall x\in K\setminus\{0\})(\exists y\in K):xy=yx=1.$$

Answer 2

For $GF$ to be a field it has to have for every element (except for $0$) an multiplicative inverse. Hence only $1$ and $5$ has one and the others don’t, it’s not an field. For every finite field such as it has an number of elements which is not prime $p$ many or $p^n$ for $n$ is natural, it can’t be an field.