What is the characteristic of an field extension?

Question

Does the characteristic remain unchanged when we extend a field?

For example, the $p$-adic field $\Bbb Q_p$ has characteristic $0$.

What is the characteristic of the finite extension $\Bbb Q_2(\sqrt{-3})$?

Answer 1

Yes, you actually do not need that the field extension is finite. For, if $K$ is a field extension of $k$, then, since $1 + \dots + 1$ is in $k$, we have that $1 + \dots + 1$ is zero in $K$ if and only if $1 + \cdots + 1$ is zero in $k$.

Edit: A more wordy formulation of this argument goes like this. For a field $k$, the prime subfield $F_k$ is defined to be the field generated by $1 \in k$. Note that $F_k$ has the same characteristic as $k$, since the image of $\mathbb Z \rightarrow k$ given by $n \mapsto n \cdot 1$ is contained in $F_k$. Let $K$ be a field extension of $k$. We have $1 \in K \cap k$ (inclusion is a field homomorphism, and field homomorphisms map $1$ to $1$). So $F_K = F_k$, which implies that $K$ and $k$ have the same characteristic.

Answer 2

More generally, a ring extension of $R$ is a ring containing an isomorphic image of $R$. Note that the order of an element is a ring-theoretic object, i.e. it is preserved under ring (or group) isomorphisms; in particular the (additive) order of $\,1\,$ is preserved, i.e. the characteristic of $R$.