What finite fields are quadratically closed? A field is quadratically closed if each of its elements is a square.
The field $\mathbb{F}_2$ with two elements is obviously quadratically closed.
However, testing some more finite fields with this property, I didn't find any more. Hence my question is:

Which finite fields $\mathbb{F}_{p^n}$ are quadratically closed and
why?


$W^{\wedge}(F)$ is the Witt-Grothendieck ring of $F$, informally it can be thought of set of all formal differences of regular quadratic forms on $F$(this is not exactly right).
$F$ is quadratically closed $\Leftrightarrow$  $dim \colon W^{\wedge}(F) \to \mathbb{Z} $ is an isomorphism $ \Leftrightarrow \lbrace dim(q_{1})=dim(q_{2}) \Leftrightarrow q_{1}=q_{2} \rbrace$ 
.
Simply speaking,it just states for a given finite dimensional vector space $V$ over $F$ ,there should be exactly one quadratic form on $V$,upto isometry.
