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1)Prove that \(\displaystyle P\cdot\overline{P}=pR\)

2)Prove that if \(\displaystyle P\) is principal then \(\displaystyle p=A^2+13B^2\) for some \(\displaystyle A,B\) integers

3) Prove that if \(\displaystyle p=a^2+13b^2\) then \(\displaystyle P\) is principal

4) Deduce that \(\displaystyle \mathbb{Z}[\sqrt{-13}]\) is not a PID

I have a proof of point 1). Indeed it is easily seen that \(\displaystyle P\cdot\overline{P}\) can be generated by \(\displaystyle p^2,p(a\pm b\sqrt{-13}),a^2+13b^2\), all elements of \(\displaystyle pR\). Conversely, i proved that we may always suppose, without loss of generality, that \(\displaystyle p^2\) does not divide \(\displaystyle a^2+13b^2\) hence \(\displaystyle p=\gcd(p^2,a^2+13b^2)\), and this last fact implies that \(\displaystyle p\) can be expressed as \(\displaystyle \mathbb{Z}\)-linear combination of \(\displaystyle p^2\) and \(\displaystyle a^2+13b^2\), both elements of \(\displaystyle P\cdot\overline{P}\) so that we get the inclusion \(\displaystyle pR\subseteq P\cdot\overline{P}\).

For the other points, any help would be appreciated