[Math] Does a field extension $L$ of $F$ have the same characteristic as $F$

Question

Is the following proof sound? Is it true for all field extension finite and infinite?

Suppose $F$ has characteristic $p$. Since $1_l=1_f$ this implies that $p*1_f=p*1_l=0$. $\text{char}(L)=p_1$ cannot be less than $p$ as this implies that $p_1*1_f=0$. Hence, $\text{char}(L)=p$. Similarly if $F$ has characteristic $0$ this implies that $L$ cant have finite characteristic (same reasoning as above.)

Answer 1

If $L$ is a field extension of $K$, then $K$ is additively a subgroup of the additive group of $L$. This inherits the characteristic.