Is $\text{Frac}(O_K)=K$? Let $K$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ with ring of integers $O_K$.
We know that $\mathbb{Q}_p$ is the field of fractions of $p$-adic integers $\mathbb{Z}_p$ i.e., $$\mathbb{Q}_p=\text{Frac}(\mathbb{Z}_p).$$
Similarly, $O_K$ is also an integral domain with no zero divisor. So we can take field of fractions of $O_K$.
My question:
Is $\text{Frac}(O_K)=K$ ?
I believe so.
any help please

Let $x\in K$. Then we have $x\in\mathcal{O}$, if $|x|\leq 1$. Otherwise $x^{-1}\in\mathcal{O}_{K}$, since $|x^{-1}| = |x|^{-1} < 1$ for $|x| > 1$. So $x\in\mathrm{Frac}(\mathcal{O}_{K})$ and hence $K\subset\mathrm{Frac}(\mathcal{O}_{K})$.
