I was recently reading about fields like $\mathbb {Z}_p$ and I'm wondering what's the reason they can't be the same element.
Is it about the additive identity being the only element with no multiplicative inverse? To be honest it's the only thing that comes to mind but it still doesn't tell me why it should be like that.
Best Answer
It is possible to define a field with just one element, which has to be the additive and multiplicative identity at the same time. Most definitions exclude this from being a field. If you have at least two elements in your field and try to make the identities the same you fail. Call the common identity $0$ and the other element $a$. But then $$0 \cdot a=a\\(0-0)\cdot a=a\\0\cdot a - 0 \cdot a=a\\a-a=a\\0=a$$