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Let $\alpha:=\sqrt[3]{7}$ and let $K:=\mathbb{Q}[\sqrt[3]{7}]$. Consider a generic algebraic integer $a+b\alpha+c\alpha^2$, with $a,b,c\in\mathbb{Z}$. I want to find $N(a+b\alpha+c\alpha^2)$, where $N$ denotes the norm of $K$ over $\mathbb{Q}$. I wrote $N(a+b\alpha+c\alpha^2)=(a+b\alpha+c\alpha^2)(a+b\alpha\zeta+c\alpha^2\zeta^2)(a+b\alpha\zeta^2+c\alpha^2\zeta)$, where $\zeta$ is a 3rd primitive root of $1$, and after a very very long computation I found $a^3+7b^3+49c^3-21abc$.

Is there some way to shorten this?

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I personally don’t think that’s much of calculation :), but there are other ways to find the norm.

Namely, let $\beta=a+b\alpha+c\alpha^2$. We want to find the matrix of the multiplication map $M_\beta$. Well,

$$\beta\cdot 1=\beta=a+b\alpha+c\alpha^2$$

$$\beta\alpha=a\alpha+b\alpha^2+c\alpha^3=7c+a\alpha+b\alpha^2$$

$$\beta \alpha^2=a\alpha^2+b\alpha^3+c\alpha^4=7b+7c\alpha+a$$

So,

$$M_\beta=\begin{pmatrix} a & b & c\\ 7c & a & b\\ 7 b & 7c & a\end{pmatrix}$$

and thus

$$N_{K/\mathbb{Q}}(\beta)=\det M_\beta=a^3-21abc+7b^3+49b^3$$

In practice, if you wanted to find $N_{L/K}(x)$ for some element $x\in L$, you can use particular properties about $x$. For example, $N_{L/K}(x)=(-1)^n a_0^{\frac{n}{d}}$ where $a_0$ is the constant coefficient of the minimal polynomial of $x$ over $K$, and $d=[K(x):K]$. Or, more generally to try and reduce your calculation to a smaller subextension (the last formula is a teased out version of this, where you restrict to the subextension $K(x)/K$).

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