Results in algebraic number theory regarding ramified split and inert primes in quadratic fields I am currently reading some notes in algebraic number theory but they are not really self contained and I am guessing the following results must hold. Let $K$ be a quadratic field and consider the ring $\mathcal{O}_K$ If $p$ is ramified in $K$ then there is a unique prime ideal of norm a power of $p$ and this ideal has norm $p$ If $p$ is split in $K$ there are exactly two ideals of norm a power of $p$ and they have norm $p$ If $p$ is inert in $K$ then there are no prime ideals of norm $p$ Of course by definition of ramified and split it is clear that there exist an ideal of norm $p$ in the case $p$ is ramified and that there exists two ideals of norm $p$ in the case $p$ is split. What is not clear to me is why those are the only ideals of norm a power $p$? Thanks in advance

Let $\mathfrak p$ be an ideal of norm $p^k$ for some $k$. Since $\mathfrak p$ is a prime ideal, $(q) = \mathfrak p\cap \mathbb Z$ must be a prime ideal. The map $$\mathbb Z\hookrightarrow\mathcal O_K\to\mathcal O_K/\mathfrak p$$ has kernel $\mathbb Z\cap\mathcal p$, so $\mathbb Z/(q)$ is a subfield of $\mathcal O_K/\mathfrak p$.

But, by the definition of the ideal norm, $$N\mathfrak p = \\#\mathcal O_K/\mathfrak p=p^k,$$ from which it follows that $p=q$.

Hence, $\mathfrak p$ lies over $p$, so there can be no other primes with norm a power of $p$.