Prove that every sum of squares in $K$ is a square in $K$, where $K$ is certain field.

Question

Let $K$ be a field such that $f(t)=t^{2}+1$ is an irreducible polynomial in $K[t]$. Let $i$ be a root of $f$ in an algebraic closure of $K$. Suppose every element of $K(i)$ is a square in $K(i)$. Prove that every sum of squares in $K$ is a square in $K$.