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.

Answer 2

This is correct. To be a bit nit-picky, you should really think of $p * 1$ as $\underbrace{1+1+1+\dots+1}_{p\text{ times}}$ and not as a multiplication in the field.

Answer 3

As an alternative way to see this, a field $k$ has characteristic $p$ precisely if it contains $\mathbb F_p$ (and characteristic $0$ if and only if it contains $\mathbb Q$). This is proved by considering the unique ring map $\mathbb Z\to k$, which has the property that the unique nonnegative generator of its kernel is the characteristic of $k$. Then one can factor this map uniquely through $\mathbb F_p$ (or $\mathbb Q$).