# Technique for showing an element is not in a field?

I have an extension $\mathbb{Q}(5^{1/4}, i)$, and I want to show that $4^{1/4}$ is not contained in it.

(I hope what I am trying to prove is true!)

Anyways, my natural starting point is to assume for a contradiction that there exist polynomials $p(x,y)$ and $q(x,y)$ both over $\mathbb{Q}$ such that $4^{1/4} = \frac{p(5^{1/4},i)}{q(5^{1/4},i)}$.

But I have no clue what to do from here, yet I suspect there is a standard technique for showing things like this? (assuming it's true)

Edit:

I have an extension, which I want to show is not normal, so I want to show that it contains $\mathbb{Q}(5^{1/4}, i)$, which is not normal over $\mathbb{Q}$, by the following argument:

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