I am thinking about the following problem: Let $\mathbb{Q}(t)$ be the field of rational functions in one variable over $\mathbb{Q}$. Consider the intermediate field extension $\mathbb{Q}(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3})$ of $\mathbb{Q}$ in the algebraic closure $\overline{\mathbb{Q}(t)}$. I would like to find out whether $\mathbb{Q}(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3}) \cap \overline{\mathbb{Q}} = \mathbb{Q}$, i.e. whether $\mathbb{Q}(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3})$ contains no proper algebraic field extension of $\mathbb{Q}$.

My thoughts so far: The extension $\mathbb{Q}(\sqrt{2}+t^{1/2}) \mid \mathbb{Q}$ is transcendental and hence $\mathbb{Q}(\sqrt{2}+t^{1/2})$ does not contain any element of $\overline{\mathbb{Q}}\setminus\mathbb{Q}$. However, $\mathbb{Q}(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3})$ is algebraic over $\mathbb{Q}(\sqrt{2}+t^{1/2})$ as they are both subfields of $\overline{\mathbb{Q}(t)}$, which has transcendence degree $1$ over $\mathbb{Q}$.

My guess is that $\mathbb{Q}(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3})$ does not contain an element algebraic over $\mathbb{Q}$ which is not already contained in $\mathbb{Q}$. Unfortunately my only approach so far is "brute force", i.e. to assume that there exist polynomials $p,q$ with integer coefficients in two variable such that $\frac{p(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3})}{q(\sqrt{2}+t^{1/2},\sqrt{3}+t^{1/3})} \in \overline{\mathbb{Q}}\setminus\mathbb{Q}$ and to calculate the resulting coefficients to derive a contradiction. But this becomes messy very quickly.

Does anyone have any suggestion what I could try otherwise?


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