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Unclear passage in J. Silverman's ATEC, Main Theorem of CM

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?

Hair80
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