Let $L/K$ be a finite field extension, and let $\{b_1,b_2,…,b_d\}$ be a basis for $L/K$
My notes define
$O_k:=\mathbb{B}\cap K$, where $\mathbb{B}:=\{\alpha$ is algebraic|min poly of $\alpha$ over $K$ has integer coefficients$\}$
Note:$\mathbb{Z}\subseteq O_k$
$Nm_{L/K}(\alpha)=Nm(m_\alpha)=$ Norm of matrix $m_\alpha$
$Tr_{L/K}(\alpha)=Tr(m_\alpha)=$ Trace of matrix $m_\alpha$
Where the $i^{th}$ column in $m_\alpha$ is the coefficients of $\alpha b_i$ as a linear expression of the basis for $L/K$.
My question is, if $Nm(\alpha),Tr(\alpha)\in \mathbb{Z}$, then does that mean that $\alpha\in O_k$, i.e the minimum polynomial of $\alpha$ over $K$ has coefficients in $\mathbb{Z}$
Best Answer
The norm and trace are just two coefficients of the characteristic polynomial. When $d\geq 3$, there is no reason that their integrality implies integrality of the polynomial.
For example, take: $$\alpha = \frac{\sqrt{1+\sqrt{17}}}{2}$$ with $K=\mathbb{Q}$, $L=\mathbb{Q}(\alpha)$. $\alpha$ has trace $0$ and norm $-1$, but is not integral (it has minimal polynomial $X^4 - \frac{1}{2}X^2 - 1$).