this problem appear in book : Karpilovsky , Unit Groups of Classical Rings. i don't know how i can solve this. the only way comes to my mind is : if we can prove $E$ has a proper subfield F such that $[E:F]<\infty$ then we can use theorem Artin-schreier to prove the result.
theorem Artin-scherier : Let $E$ be an algebraically closed field and $R$ a proper subfield of $E$ such that $[E:R]<\infty$ then $R$ is real cloesd and $E = R(\sqrt{-1})$
sorry for bad english!