Let $f=X^3-3\in\mathbb{Z}$. Suppose $E_f/k$ is an extension, where $k$ is a field. I want to find $E_f$ such that $f$ decomposes in $E_f[X]$ and $E_f=k[a_1,...,a_n]$, where $a_1,...,a_n$ are the roots of $f\in E[X]$.
How should I approach if $k=\mathbb{Q}$? The roots of $f$ in $\mathbb{C}$ are $\sqrt[3]{3},-\frac{\sqrt[3]{3}}{2}+i\frac{3^{\frac{5}{5}}}{2},-\frac{\sqrt[3]{3}}{2}-i\frac{3^{\frac{5}{5}}}{2}$. And so, I may take $E_f=\mathbb{Q}\left[\sqrt[3]{3},-\frac{\sqrt[3]{3}}{2}+i\frac{3^{\frac{5}{5}}}{2},-\frac{\sqrt[3]{3}}{2}-i\frac{3^{\frac{5}{5}}}{2}\right]$, correct?
If $k=\mathbb{F}_3$, I may take $E_f=\mathbb{F}_3$ because $f$ becomes $X^3$ and it is already factored. Is this correct?
If $k=\mathbb{F}_5$, then I don't really know how to proceed.
But if, for example, $k=\mathbb{F}_2$ and $f=X^3+X+1$, then I take $\alpha=X+(f(X))\in\mathbb{F}_2[X]/(f(X))$. And so, $\mathbb{F}_2(\alpha)$ is a field, $\alpha^3=X+1$ and $f=(X+\alpha)(X^2+aX+b)$. I find that $a=\alpha, b=1+\alpha$. I list the elements $c\alpha^2+d\alpha +e$ of $\mathbb{F}_2(\alpha)$ ($c,d,e\in\mathbb{F}_2$) because they are just $8$. Finally, I note that $\alpha^2,1+\alpha^2\in\mathbb{F}_2(\alpha)$ are roots of $X^2+\alpha X+1+\alpha^2$. Hence, $f$ splits in $\mathbb{F}_2(\alpha)$ and I take $E_f=\mathbb{F}_2(\alpha)$.
Now, I don't think this method is very useful for the third example because we would have to check $5^3=125$ elements of $\mathbb{F}_5(\alpha)$. Is there a quicker way to solve this?
I appreciate any help. Many thanks in advance!
3 already has a cube root in $\mathbb{F}_5$, so $X^3 - 3$ can be divided by a linear polynomial over $\mathbb{F}_5$ to give a quadratic polynomial, and then you just need a quadratic extension to split it.