1. ## Minimal Polynomial

Without finding the minimal polynomial for $r=i\cdot 2^{\frac{1}{3}}+i$, show the degree of said polynomial must be $6$. I know this is true because I was able to derive $r\text{'s}$ minimal polynomial ( $f(x)=x^6+3x^4-9x^2+9$), but in doing so I used the assumption that it was of degree $6$ as opposed to $2$ or $3$.

2. Originally Posted by chiph588@
Without finding the minimal polynomial for $r=i\cdot 2^{\frac{1}{3}}+i$, show the degree of said polynomial must be $6$. I know this is true because I was able to derive $r\text{'s}$ minimal polynomial ( $f(x)=x^6+3x^4-9x^2+9$), but in doing so I used the assumption that it was of degree $6$ as opposed to $2$ or $3$.
show that both $i$ and $\sqrt[3]{2}$ are in $\mathbb{Q}(i\sqrt[3]{2} + i)$ and thus $\mathbb{Q}(i\sqrt[3]{2} + i)=\mathbb{Q}(\sqrt[3]{2},i)$. we also have $[\mathbb{Q}(\sqrt[3]{2},i):\mathbb{Q}]=[\mathbb{Q}(\sqrt[3]{2},i):\mathbb{Q}(\sqrt[3]{2})] \times [\mathbb{Q}(\sqrt[3]{2}): \mathbb{Q}]=2 \times 3 = 6.$

3. Originally Posted by NonCommAlg
show that both $i$ and $\sqrt[3]{2}$ are in $\mathbb{Q}(i\sqrt[3]{2} + i)$ and thus $\mathbb{Q}(i\sqrt[3]{2} + i)=\mathbb{Q}(\sqrt[3]{2},i)$. we also have $[\mathbb{Q}(\sqrt[3]{2},i):\mathbb{Q}]=[\mathbb{Q}(\sqrt[3]{2},i):\mathbb{Q}(\sqrt[3]{2})] \times [\mathbb{Q}(\sqrt[3]{2}): \mathbb{Q}]=2 \times 3 = 6.$
Let $\alpha=i\sqrt[3]{2}+i$.

It turns out $\sqrt[3]{2} = -\frac{1}{6}\alpha^4-\alpha^2+\frac{1}{2}$
and $i = \frac{1}{6}\alpha^5+\frac{2}{3}\alpha^3-\frac{1}{2}\alpha$,
so indeed we know $\mathbb{Q}(\sqrt[3]{2},i) \subseteq \mathbb{Q}(\alpha)$.

But is there and easier way to show $\sqrt[3]{2},i\in \mathbb{Q}(\alpha)$?

4. Originally Posted by chiph588@
Let $\alpha=i\sqrt[3]{2}+i$.

It turns out $\sqrt[3]{2} = -\frac{1}{6}\alpha^4-\alpha^2+\frac{1}{2}$
and $i = \frac{1}{6}\alpha^5+\frac{2}{3}\alpha^3-\frac{1}{2}\alpha$,
so indeed we know $\mathbb{Q}(\sqrt[3]{2},i) \subseteq \mathbb{Q}(\alpha)$.

But is there and easier way to show $\sqrt[3]{2},i\in \mathbb{Q}(\alpha)$?
well, this is how i did it: $\alpha^3=-3i(1+\sqrt[3]{2} + \sqrt[3]{4})=-3 \alpha - 3i \sqrt[3]{4}$ and so $\beta=i \sqrt[3]{4} \in \mathbb{Q}(\alpha)$ and we're done because $\sqrt[3]{2}=\frac{-\beta^2}{2}$ and $i=\frac{-\beta^3}{4}.$