additive identity of a subfield

Question

Let $F$ be a field and $K$ a subfield(a subset of $F$ which is a field.) Since $F,K$ are groups and $K\subset F$, $K$ is a subgroup. For the additive identities $0_F,0_K$, we have $0_F+0_K=0_K=0_K+0_K$, so $0_F=0_K$. I was wondering if the above is true.