Let $\mathbb{Q}$ be the field of rational number. Suppose that $\alpha$ is a root of an irreducible polynomial $f(x)$ of degree $n$ in $\mathbb{Q}[x]$. Then $\mathbb{Q}[\alpha]$ is the field extension of $\mathbb{Q}$ and $\mathbb{Q}[\alpha]\cong \mathbb{Q}[x]/(f(x))$.
My question is: How can I represent an arbitrary element of $\mathbb{Q}[\alpha]$ in term of $\alpha$ and some elements in $\mathbb{Q}$?
Is : $\mathbb{Q}[\alpha]=\lbrace a_{0}+a_1\alpha+...+a_{n-1}\alpha^{n-1}|\ \text{for some}\ a_{i}\in \mathbb{Q} \rbrace$?
I have a concrete problem with $\mathbb{Q}[\sqrt[3]{2}\zeta]$ where $\zeta \neq 1$ is the third root of unity, I do not know how to represent exactly an arbitrary element of $\mathbb{Q}[\sqrt[3]{2}\zeta]$ in term of elements in $\mathbb{Q}$ and $\sqrt[3]{2}, \zeta$.
Please help me.
Thanks.
Yes your element description of $\mathbb{Q}[\alpha]$ is correct. For $\alpha = 2^{1/3} \cdot \zeta_3$ we have $\alpha^3=2$, and hence $f=x^3-2$ (this is irreducible by Eisenstein). Thus $\mathbb{Q}[\alpha]$ has as $\mathbb{Q}$-basis $1,\alpha,\alpha^2$.