Suppose $K$ is a field and $F$ is a subfield of $K$. Let $1_K$ be the multiplicative identity of $K$ and let $1_F$ be the multiplicative identity of $F$. Is it true that $1_K = 1_F$?
If the hypothesis that $K$ and $F$ are fields were removed and if we require only that $K$ is a ring and $F$ is a subring of $K$, then it is possible that $1_K$ and $1_F$ both exists but $1_K = 1_F$ is not true. For example, let $K = \mathbb{Z}/6\mathbb{Z}$ and let $F = \{0, 2, 4\}$ where $0$, $2$, and $4$ are elements in $\mathbb{Z}/6\mathbb{Z}$ and the addition and the multiplication of $F$ are inherited from $K$. Then $1$ is the multiplicative identity of $K$ and $4$ is the multiplicative identity of $F$.
I can't prove that there is no such case if $K$ is a field and $F$ is a subfield of $K$.
The roots of $x^2-x$ in $K$ are $0_K$ and $1_K$.
$1_F$ is also a root of $x^2-x$ in $K$ because $F$ is a subring of $K$.
If $1_F=0_K$, then $a=a1_F=a0_K=0$ for all $a \in F$, and $F$ cannot be a subfield.
Therefore, $1_F=1_K$.