I am reading through the section of Atiyah-MacDonald on fractional ideals.
They describe the group of invertible fractional ideals for an integral domain $A$ in its field of fractions $K$.
They then go on to discuss the situation where you consider a number field $K$ and its ring of integers $O_K$.
Is the field of fractions of $O_K$ isomorphic to $K$, or a field containing/contained in $K$? Is this true more generally (without all of these Dedekind domains floating around)?
On
Let $K$ be a field and $R\subset K$ a subring. Consider $R_K'$ the ring of integers of $R$ in $K$, that is, $R'_K=\{x\in K:x \text{ is integral over } R\}$. Then the field of fractions of $R'_K$ is contained in $K$ since we know that the field of fractions of an integral domain is the smallest field containing it.
For example, let $R=\mathbb Z$, and $K=\mathbb R$. Then the field of fractions of $R'_K$ is the field of algebraic numbers.
However, in the particular case when $K$ is an algebraic extension of the field of fractions of $R$, we have that the field of fractions of $R'_K$ is $K$. Let $x\in K$. Then $x$ is algebraic over the field of fractions of $R'_K$, so there is $b\in R$ such that $bx$ is integral over $R$, that is, $bx\in R'_K$.
In the case of a number field $K$ where $\mathcal{O}_K$ is the integral closure of $\mathbb{Z}$ (I hope that's what you mean since in this case $K$ is the field of fractions of $\mathcal{O}_k$) we have the groups $I_K$ and $P_K$. $I_K$ are all fractional ideals of $K$ and $P_K$ only the principal fractional ideals. Since we have the surjective group homomorphism $$\varphi:K^\times \to P_K, \quad \alpha \mapsto (\alpha)$$ we clearly see that $K^\times /\ker(\varphi) \cong P_K$. Since two principal ideals are the same iff the generators are associated we see $\ker(\varphi)=\mathcal{O}_K^\times$. But we always have $-1\in\mathcal{O}_K^\times$, so $\ker(\varphi)$ is never trivial and therefore $K^\times$ is never isomorphic to $P_K$.