Reading the proof of the Main Theorem of CM for elliptic curves over number fields in J. Silverman's book "Advanced Topics of Elliptic Curves" I got stuck at a passage which looks quite innocuous but which I can not provide a reasonable proof about. On page 126, proving Proposition 4.2 the following statement is given. Let $p$ be a prime ideal in $\mathbb{Z}$ which splits completely in the quadratic field extension $K$, so that $pO_{K}=\mathfrak{p}\mathfrak{q}$. Then it is always possible to find an ideal $\mathfrak{a}$ in $O_{K}$, coprime with $pO_{K}$ and such that $\mathfrak{a}\mathfrak{p}$ is principal. I do not see any immediate reason why this should be true? I have found a paper by K. Conrad stating that for any $\mathfrak{a}$ in $O_{K}$ then if $\overline{\mathfrak{a}}$ is the conjugate of $\mathfrak{a}$ (the ideal of all conjugates of the elements of $\mathfrak{a}$), we have that $\mathfrak{a}\overline{\mathfrak{a}}$ is principal. This seems weaker than the statement in J. Silverman's book and still very complicated to prove. Is there any simple proof I am missing?

Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage which looks quite innocuous but which I can not provide a reasonable proof about. On page 126, proving Proposition 4.2 the following statement is given. Let $p$ be a prime ideal in $\mathbb{Z}$ which splits completely in the quadratic field extension $K$, so that $pO_{K}=\mathfrak{p}\mathfrak{q}$. Then it is always possible to find an ideal $\mathfrak{a}$ in $O_{K}$, coprime with $pO_{K}$ and such that $\mathfrak{a}\mathfrak{p}$ is principal. I do not see any immediate reason why this should be true? I have found a paper by K. Conrad stating that for any $\mathfrak{a}$ in $O_{K}$ then if $\overline{\mathfrak{a}}$ is the conjugate of $\mathfrak{a}$ (the ideal of all conjugates of the elements of $\mathfrak{a}$), we have that $\mathfrak{a}\overline{\mathfrak{a}}$ is principal. This seems weaker than the statement in J. Silverman's book and still very complicated to prove. Is there any simple proof I am missing?

Reading the proof of the Main Theorem of CM for elliptic curves over number fields in J. Silverman's book "Advanced Topics of Elliptic Curves" I got stuck at a passage which looks quite innocuous but which I can not provide a reasonable proof about. On page 126, proving Proposition 4.2 the following statement is given. Let $p$ be a prime ideal in $\mathbb{Z}$ which splits completely in the quadratic field extension $K$. Then it is always possible to find an ideal $\mathfrak{a}$ in $O_{K}$, coprime with $pO_{K}$ and such that $\mathfrak{a}(pO_{K})$ is principal. I do not see any immediate reason why this should be true? I have found a paper by K. Conrad stating that for any $\mathfrak{a}$ in $O_{K}$ then if $\overline{\mathfrak{a}}$ is the conjugate of $\mathfrak{a}$ (the ideal of all conjugates of the elements of $\mathfrak{a}$), we have that $\mathfrak{a}\overline{\mathfrak{a}}$ is principal. This seems weaker than the statement in J. Silverman's book and still very complicated to prove. Is there any simple proof I am missing?

Reading the proof of the Main Theorem of CM for elliptic curves over number fields in J. Silverman's book "Advanced Topics of Elliptic Curves" I got stuck at a passage which looks quite innocuous but which I can not provide a reasonable proof about. On page 126, proving Proposition 4.2 the following statement is given. Let $p$ be a prime ideal in $\mathbb{Z}$ which splits completely in the quadratic field extension $K$, so that $pO_{K}=\mathfrak{p}\mathfrak{q}$. Then it is always possible to find an ideal $\mathfrak{a}$ in $O_{K}$, coprime with $pO_{K}$ and such that $\mathfrak{a}\mathfrak{p}$ is principal. I do not see any immediate reason why this should be true? I have found a paper by K. Conrad stating that for any $\mathfrak{a}$ in $O_{K}$ then if $\overline{\mathfrak{a}}$ is the conjugate of $\mathfrak{a}$ (the ideal of all conjugates of the elements of $\mathfrak{a}$), we have that $\mathfrak{a}\overline{\mathfrak{a}}$ is principal. This seems weaker than the statement in J. Silverman's book and still very complicated to prove. Is there any simple proof I am missing?