# example of $L/\Bbb{Q}_p$ such that there is no prime element $π$ of ring of integers $L$ such that $p＝π^e$.

Let $$L/\Bbb{Q}_p$$ be ramification index $$e$$ extension. Let $$π$$ be prime element of $$L$$. Then, $$p＝π^eu$$ ($$u$$: unit element of ring of integers of $$L$$)。

But I have met the remark that we can choose prime element $$π$$ such that $$p＝π^e$$ (I cannot find the reference now, sorry・・・).

I think this is true in the case $$e＝1$$, but when $$e≧2$$, I think we cannot take $$π$$ in general・・・

Is the remark correct statement ? If not, I want to know counterexample, that is, example of $$L/\Bbb{Q}_p$$ such that there is no prime element $$π$$ such that $$p＝π^e$$.

• Try with $L=\Bbb{Q}_2(1+i)$ Feb 25, 2022 at 11:01
• In the case $e＝1$, the remark is right, but when $e≧2$ the remark does not hold, is that correct ?
– Pont
Feb 25, 2022 at 11:07
• Why do you think that $e=1$ Feb 25, 2022 at 11:08
• I'm not assuming $e＝1$. In the case of $e＝1$, the remark is correct, I think. The matter is when $e≧2$.
– Pont
Feb 25, 2022 at 11:11
• Try $L=\Bbb{Q}_2(1+i)$ and see what you get Feb 25, 2022 at 11:11

Let $$d$$ be a quadratic nonresidue modulo $$p$$, and take $$L = \mathbb{Q}_p(\sqrt{dp})$$. Then $$e = 2$$, so we want to show that $$p$$ is not square in $$L$$.

Suppose that $$p$$ is square in $$L$$. Then there is some $$\alpha \in L$$ with $$p = \alpha^2$$. Clearly there are $$a,b\in\mathbb{Q}_p$$ such that $$\alpha = a + b\sqrt{dp}$$, so $$p = (a + b\sqrt{dp})^2 = a^2 + b^2dp + 2ab\sqrt{dp},$$ so we have $$p = a^2 + b^2dp, \quad 2ab = 0.$$ Since $$2ab = 0$$, either $$a = 0$$ or $$b=0$$. Suppose that $$a = 0$$. Then $$b^2dp = p$$ so $$b^2d = 1$$. This implies that $$\lvert b^2\rvert_p = 1$$, so $$b \in \mathbb{Z}_p^\times$$ and $$d \equiv (b^{-1})^2 \pmod{p}$$, which is impossible since $$d$$ is a quadratic nonresidue. Suppose instead that $$b = 0$$. Then $$a^2 = p$$, which is impossible since $$a \in \mathbb{Q}_p$$.

• Yes, indeed. There do exist non-square units $u$, and then $\sqrt{up}$ generates a ramified quadratic extension of $\mathbb Q_p$ in which $p$ is not a square. Jul 20, 2022 at 20:04