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Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois group of $\mathbb{Q}(\zeta)/\mathbb{Q}$. $G$ is isomorphic to $(\mathbb{Z}/l\mathbb{Z})^*$. Hence $G$ is a cyclic group of order $l - 1$. Let $f = (l - 1)/2$. There exists a unique subgroup $G_f$ of $G$ whose order is $f$. Let $K_f$ be the fixed subfield of $K$ by $G_f$. $K_f$ is a unique quadratic subfield of $K$. Let $A_f$ be the ring of algebraic integers in $K_f$.

Let $p$ be a prime number such that $p \neq l$. Let $pA_f = P_1\cdots P_r$, where $P_1, \dots, P_r$ are distinct prime ideals of $A_f$.

Since $p^{l - 1} \equiv 1$ (mod $l$), $p^f = p^{(l - 1)/2} \equiv \pm$1 (mod $l$).

My question: Is the following proposition true? If yes, how would you prove this?

Proposition

(1) If $p^{(l - 1)/2} \equiv 1$ (mod $l$), then $r = 2$.

(2) If $p^{(l - 1)/2} \equiv -1$ (mod $l$), then $r = 1$.

This is a related question.

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Makoto Kato
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    You seem to like cyclotomic stuff a lot – Asinomás Jul 24 '12 at 23:24
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    Because I believe they are at the heart of algebraic number theory. – Makoto Kato Jul 24 '12 at 23:26
  • I think you should invest in a good reference for algebraic number theory like Neukirch. Many of your recent questions are standard cyclotomic ideas that are often addressed. – bzc Jul 24 '12 at 23:38
  • @BrandonCarter Does the Neukirch's book have the following result? http://math.stackexchange.com/questions/174703/decomposition-of-a-prime-number-p-in-a-subfield-of-a-cyclotomic-number-field-o – Makoto Kato Jul 24 '12 at 23:53
  • @Makoto: Not to my knowledge, which is why I said many of your recent questions and not all. – bzc Jul 25 '12 at 04:04
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    Dear Makoto, It may not state precisely that result, but it will certainly establish (much more than) the tools necessary to prove it, which is just the theory of decomposition groups. Neukirch's book is not my own favourite, but any book (or online set of lecture notes) on algebraic number theory will develop these tools. Since they are applicable to many of your questions, you may well want to learn them. Regards, – Matt E Jul 26 '12 at 03:28
  • @MattE Dear Matt, If someone asks a question which can be answered by using the above result, I think it's easier and perhaps better to show him the link(under the condition that it has a good answer) than to write the result and say this can be proved by the notion of decomposition groups and refer to Neukirch, etc.. Regards, – Makoto Kato Jul 26 '12 at 04:10

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This is most easily established by decomposition group calculations; it is a special case of the more general result proved here (which is an answer to OP's linked question).

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