Is there a normal extension $L$ such that $\mathbb Q \subset \mathbb Q(\sqrt3) \subset L$ with cyclic $\text{Gal}(L/\mathbb Q) \cong \mathbb Z_4^+$

Question

I am just beginning Galois and don't have much control over the material.

Is there a normal extension $L$ such that $\mathbb Q \subset \mathbb Q(\sqrt3) \subset L$ with cyclic $\text{Gal}(L/\mathbb Q) \cong \mathbb Z_4^+$.

1. I already know that $\text{Gal}(\mathbb Q(\sqrt3)/\mathbb Q)$ consists the identity and the automorphism such that $\mp \sqrt3 \mapsto \pm \sqrt3$.
2. With $\text{Gal}(L/\mathbb Q) \cong \mathbb Z_5^\times \cong \mathbb Z_4^+$, we have one permutation of order $1$, one of order $2$, and two of order $4$.

So, these clues suggest that $L = \mathbb Q(\sqrt3, i)$. But according to this, such an extension does not exist. Where did I make a mistake?