Cardinality of the torsion subgroup of a field.

Question

In an exercise I encountered the following question:

Let $k$ be a field, what is the cardinality of the torsion subgroup of $k$?

Let distinguish the following cases:

• If $\textrm{char}(k)=0$, then $k_{\textrm{tor}}=\{0\}$. Indeed, if $x\in k^\times$, then for all $n\in\mathbb{N}$, $nx=0\implies n=0$. Whence: $$\#k_{\textrm{tor}}=1.$$

• If $\textrm{char}(k)=p$, then if $k$ is a finite field using Lagrange's theorem, one has $k_{\textrm{tor}}=k$. Whence: $$\#k_{\textrm{tor}}=\#k.$$ When $k$ is a infinite field of characteristic $p$, e.g. $k=\mathbb{F}_p(X)$, I have no clue on how to proceed.

Any help would be greatly appreciated.

Answer 1

To sum up your work and the remainder in the comments:

Let $k$ be a field of characteristic $n$.

• If $n=0$ then for all $x\in k^{\times}$ we have $mx=0$ if and only if $m=0$, so no nonzero element of $k$ is torsion, i.e. $k_{\text{tor}}=\{0\}$
• If $n>0$ then for all $x\in k$ we have $nx=0$, so every element of $k$ is torsion, i.e. $k_{\text{tor}}=k$.