Problem Understanding a Definition of Order of a Field

Question

I need some help regarding some definitons. I was studying algebraic number theory and I am stuck on this.Can someone explain me what is meant by $F = [\mathcal{O}_k:\mathcal{O}]$ with an easy examples

I was studying the following theorem but didn't get examples

Let $\mathcal{O}$ be an order in $K$. Then $F = [\mathcal{O}_k:\mathcal{O}]$ is finite and $\mathcal{O} = \mathbb{Z} + F \mathcal{O}_k$.

Moreover, any set $\mathcal{O} = \mathbb{Z} + F \mathcal{O}_k$. with $1 \leq F$ is an order in $K$ such that $F = [\mathcal{O}_k:\mathcal{O}]$

Answer 1

Since you mentioned that you're studying algebraic number theory, I'm assuming $K$ is an algebraic number field in your context.

An order $\mathcal{O}$ of a number field $K$ is defined to be a subring of the ring of integers $\mathcal{O}_K$ with $d$ generators over $\mathbb{Z}$, where $d$ is the degree of the number field.

Thus, $F = [\mathcal{O}_K : \mathcal{O}]$ simply means the index of the subring $\mathcal{O}$ in $\mathcal{O}_K$.

For example, if we are working with the number field $K = \mathbb{Q}(\alpha)$ where $\alpha^2 + 1 = 0$. One can show that the ring of integers is $\mathcal{O}_K = \mathbb{Z}[\alpha]$. Choose your favorite integer $z \in \mathbb{Z}$. Note that $\mathcal{O} = \mathbb{Z}[z\alpha] = 1\mathbb{Z} + z\alpha\mathbb{Z}$ is a subring of $\mathcal{O}_K$. Hence, $\mathcal{O}$ is an order of the number field $K$. In this case, $F = [\mathbb{Z}[\alpha] : \mathbb{Z}[z\alpha]] = z$.

You might also want to edit your question to make it more precise.

Answer 2

A simple example is $K=\mathbb Q(\sqrt d)$ and $\mathcal O=\mathbb Z[\sqrt d]$ when $\mathcal O \ne \mathcal O_K$. Take for instance $d=5$.