"By some other method" is quite vague. If, by modding out, you are thinking of taking a subring and then modding that ring by a maximal ideal (i.e., you obtain $\mathbb{Z}/p\mathbb{Z}$ from the rationals in this way), then it cannot be done.
Call the field $F$, the subring $R$, and the maximal ideal $I$, and assume $F$ has finite characteristic $p$. By definition of subring, the unit of the $R$ is also the unit of $F$. If the field has characteristic $p$, then $p \cdot 1 = 0$ is also true in the subring, and hence, in the quotient field $R/I$. Thus, the quotient field must also have characteristic $p$ if the original field does.
By definition, an extension field $K$ of $F$ would then contain $F$ as a subring, so again, they would have to have the same characteristic.