Multiplicative order of components in a fictional square area

Let $K$ be a fictional square area and also $U$ represent the device team in the ring of integers in $K$. Exist $\alpha \in K-U$ with limited multiplicative order? That is, exists $n \in N$ such that $\alpha^n=1$?

2019-12-02 03:12:25
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Answers: 1

If $\alpha^n=1$, after that $\alpha$ pleases the polynomial $x^n-1$, therefore is an algebraic integer. Hence, $\alpha\in \mathcal{O}_K$ ; as soon as you have that, you can make the very easy monitoring Qiaochu carried out in the remarks that $\alpha^{-1}=\alpha^{n-1}$ is additionally in $\mathcal{O}_K$ to show $\alpha\in U$ (or you can make use of the reality that if an algebraic integer pleases a monic polynomial with integer coefficients and also constant term $1$, after that it has to be a device). So the solution is "no."

Of training course, every origin of unity is indispensable (enjoyable $x^n-1$ for some $n$, or the ideal cyclotomic polynomial if you demand obtaining the marginal polynomial), so origins of unity in a number area are constantly in the ring of integers.

2019-12-03 05:07:13