Show that two field extensions are the same

Question

Can you help me with showing that these two field extensions are the same:

1. $\mathbb{Q}(\sqrt{3}, \sqrt[3]{5})$
2. $\mathbb{Q}(\sqrt{3} + b\sqrt[3]{5})$, where $b\neq0$ is any rational number.

Thanks

Answer 1

Let $\alpha=\sqrt 3+b\sqrt[3]5$. Clearly, $\mathbb Q(\alpha)\subseteq \mathbb Q(\sqrt 3,\sqrt[3]5)$. Let $\beta=\sqrt 3$. Then $$5b^3=(\alpha-\beta)^3=\alpha^3-3\alpha^2\beta+9\alpha-3\beta $$ shows that (NB $\alpha^2+1\ne0$) $$ \beta = \frac{\alpha^3+9\alpha-5b^3}{3(\alpha^2+1)}\in\mathbb Q(\alpha)$$ and (since $b\ne0$) then also $$ \sqrt[3]5=\frac{\alpha-\beta}{b}\in\mathbb Q(\alpha).$$