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Let $\alpha$ be an element in an extension field $K$ of $\mathbb{Q}$ generated by the cubic root of a non-cubic $u\in \mathbb{N}$ : $K=\mathbb{Q}[\sqrt[3]{u}]$, that does not lie in $\mathbb{Q}$.

My question is : What can we say about the degree of the minimum polynomial of $\alpha$ over $\mathbb{Q}$ ?

I have tried to solve this, and here is my argument :

Since $\alpha\in\mathbb{Q}[\sqrt[3]{a}]$, then $\alpha=a+b\sqrt[3]{u}+c\sqrt[3]{u^2}$, for $a,b,c \in \mathbb{Q}$.

Then $(\alpha-a)^3=u(b+c\sqrt[3]{u})^3$. Then by direct calculation we get : $\alpha^3-3a\alpha^2-3a(a-bc)\alpha-a(b^3+c^3u-3bc)=0$

Therefore the degree of the minimum polynomial of $\alpha$ over $\mathbb{Q}$ is at most 3.

What can we conclude more ?

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up vote 3 down vote accepted

You have $\mathbb{Q}(\alpha) \subset K$. Thus, $[K:\mathbb{Q}] = [K:\mathbb{Q}(\alpha)][\mathbb{Q}(\alpha):\mathbb{Q}]$. In particular, since $[K:\mathbb{Q}] = 3$, we have $[\mathbb{Q}(\alpha):\mathbb{Q}] = 1$ or $[\mathbb{Q}(\alpha):\mathbb{Q}] = 3$. Hence the degree of the minimal polynomial of $\alpha$ over $\mathbb{Q}$ must be either $1$ or $3$.

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