The variety of roots of unity in a fictional square number area

The publication I am researching makes use of the symbols $w_k$ to represent the variety of roots of unity had in $K$, a fictional square number area.

Apparently, this coincides as the dimension of the device team in $O$, the ring of integers in $K$. I am well accustomed to the typical debates for locating the devices in $O$.

Im not actually certain specifically what the expression "the variety of roots of unity in $K$" suggests specifically. Does any person recognize?

Additionally, I was assuming that it is probibly the instance that the only components of $K$ with limited multiplicative order are the components of the device team. Is this proper? If so, after that is this the proper logic to show that $w_k=|U(O)|$?

2019-12-02 03:11:16
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Answers: 2
2019-12-03 05:08:46

An origin of unity is an intricate number $a$ such that $a^n = 1$ for some favorable integer $n$. The "number of origins of unity" is the cardinality of the set of origins of unity.

As an example, if $K=\mathbb{Q}(i)$, after that the origins of unity in $K$ are $1$, $-1$, $i$, and also $-i$, so the variety of origins of unity had in $K$ is $4$.

Yes : an intricate number is an origin of unity if and also just if it has limited multiplicative order in $\mathbb{C}^*$. Actually, the team of origins of unity is the torsion subgroup of $\mathbb{C}^*$.

To confirm that in the fictional square instance the device team of $\mathcal{O}_K$ accompanies the team of origins of unity in $K$ you can make use of Dirichlet is Unit Theorem (as Timothy Wagner recommends), yet you do not need to. You can merely show that every device in $\mathcal{O}_K$ have to actually be an origin of unity. This is not hard, given that you (possibly) recognize specifically what $\mathcal{O}_K$ for $K=\mathbb{Q}(\sqrt{d})$ ($d$ square free, $d\neq 1$), and also you can make use of the standard map. As an example, to show it when it comes to $K=\mathbb{Q}(i)$, note that a component in $\mathcal{O}_K = \mathbb{Z}[i]$ is a device if and also just if $N(a+bi) = a^2+b^2 = 1$ (given that $N((a+bi)(c+di)) = N(a+bi)N(c+di)$). Yet given that $a$ and also $b$ are integers, that calls for $a=\pm 1$ and also $b=0$ or $a=0$ and also $b=\pm 1$, offering you origins of unity.

(In reality, fictional square areas are the only number areas in which the team of devices in the number area is limited and also torsion ; this adheres to from the abovementioned Dirichlet Unit Theorem)

2019-12-03 05:03:58