Is Z3 a sub-field of R?

Question

The inverse numbers for the items in $\mathbb{Z3}$ are different than in $\mathbb{R}$ so I assume it's not a sub-field of $\mathbb{R}$. Am I correct?

And in general, can sub-field of a infinite field be finite?

Answer 1

Any subfield of $\mathbb R$ is automatically of characteristic $0$ and thus contains $\mathbb Q$, in particular it is an infinite field.

Nevertheless, it is possible that a finite field is a subfield of an infinitie field, for example we have $\mathbb Z/3\mathbb Z \subset \mathbb Z/3\mathbb Z(X)$, where the latter is the function field over $\mathbb Z/3\mathbb Z$.

Answer 2

It is not, because $\mathbb Z_3$ is not really a subset of $\mathbb R$, since $\mathbb Z_3=\mathbb Z/(3\mathbb Z)$

Of course, you can say that $\mathbb Z_3$ can be defined as the set $\{0,1,2\}$, but in that case, the operation $+$ defined on the set is not the same as the operation we typically denote as $+$, since $2+2=1$ in $\mathbb Z_3$, but $2+2=4$ in $\mathbb R$.