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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 $K$.

My question: Is there a purely imaginary unit in $A$?

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

Proposition There is no purely imaginary unit in $A$.

Related questions:

On a certain property of the different of an extension of an algebraic number field of a prime relative degree

Maximal real subfield of Q(ζ)

Community
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Makoto Kato
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    Please don't edit questions in such a way that render existing answers non-sense. – Willie Wong Jul 25 '12 at 13:57
  • @WillieWong I just added a question. It does not contradict the existing question. – Makoto Kato Jul 25 '12 at 14:16
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    Don't you think that 40+ questions in 2 days is enough for the community to think about? Why not leaning back and thinking about accepting some of given answers? – draks ... Jul 25 '12 at 20:59
  • @MakotoKato absolutely true: Accept when you wish... – draks ... Jul 26 '12 at 00:18
  • @WillieWong Do you rather want me to open a new question instead of editing this? – Makoto Kato Jul 26 '12 at 02:09
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    Dear Makoto, You have been asking many questions about algebraic number theory, and in particular, the algebraic number theory of cyclotomic fields. These are very well-studied subjects, with many texts available. (E.g. the books on cyclotomic fields by Lang and Washington.) Have you considered looking at some of these books to answer your questions? Regards, – Matt E Jul 26 '12 at 02:32
  • @MakotoKato If you interpreted Matt E 's comment as buying books, I think you missed his point. As your questions seems to suggest your immense interest in algebraic number theory, you should probably engage in a systematic study of the subject and learn whatever background material is necessary. Your goal should not be to amass a great amount of facts. You should strive for understanding and the ability to solve problems. Many of your questions can be answered with sufficient understanding of number theory. You should learn in a way so that you can eventually solve problems yourself. – William Jul 26 '12 at 06:11
  • @MakotoKato If resources are an issue for you, there are quite a number of free sources on algebraic number theory. Milne has a nice book, which is available on his webpage. It covers the Dirichlet Unit Theorem which was used in one of the solutions below. Some people prefer his book over expensive published book like Neukirch. – William Jul 26 '12 at 06:14
  • @William I'm not asking questions only for myself. Good questions can be valuable in this site(I'm not necessarily implying all of my questions are good, though). Moreover, if someone asks a question which can be solved using my questions(and answers to them), anyone can use them when he/she answers it. – Makoto Kato Jul 26 '12 at 06:36
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    I'd actually say that of the OP's many recent questions on cyclotomic fields, this one (in its edited form) is probably the most interesting, so I'm not sure why people are piling in to down-vote it. – David Loeffler Jul 26 '12 at 08:46
  • @Makoto: no, I don't want you to open a new question. But I want you to think about your edits before you do them. In particular, if you first asked "Is X true?" and then changed your question to "Is it true that X is false?" this makes an answer "No" highly ambiguous. Furthermore, in this case you asked "Is X(n) true?" which can be falsified by finding an n such that X(n) is false, but changing the question to "Is it true that X(n) is false?" requires a proof that X(n) is false for all n. – Willie Wong Jul 26 '12 at 10:00
  • @WillieWong So what exactly do you want me to do? Delete the edit? – Makoto Kato Jul 26 '12 at 10:16
  • @Makoto: what is done is done; Matt E answered your question assuming your most recent edit, so modifying it would just make things even worse. In the future I want you to think before you edit. For example, it would've been better if your update was something like: "Edit: in fact, I think the answer is 'No' for any such $A$. I would appreciate a proof of that." And you may even write "Edit (in view of David Loeffler's answer):..." Basically you should avoid asking the same question as both a positive and a negative statement. Choose one and stick with it. – Willie Wong Jul 26 '12 at 10:26
  • @WillieWong I didn't ask both a positive and negative statement. I wonder why you think otherwise. – Makoto Kato Jul 26 '12 at 11:19
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    @Makoto: I can understand when you have difficulty understanding what I wrote, sometimes my use of language is a bit obtuse. But try to at least understand what you wrote yourself! I quote: positive statement "Is there a purely imaginary unit in A?" negative statement "Is the following proposition true? Proposition: there is no imaginary unit in A." – Willie Wong Jul 26 '12 at 11:44
  • @WillieWong Could you tell me your definition of "positive" and "negative" statement? – Makoto Kato Jul 26 '12 at 18:00
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    @Makoto: Given a mathematical statement $X$. A positive question is "Is $X$ true?" A negative question is "Is $X$ false?" or equivalently "Is $\neg X$ true?". Yes, my previous two comments were trying to address the same thing. – Willie Wong Jul 26 '12 at 18:34
  • @WillieWong It's not that simple. If I didn't add the latter question, I doubt that Matt E would give a definitive answer and David Loeffler would think the question interesting. Anyway, you know I almost always ask the latter type questions. I don't think this would ever happen again. I apologize anyone who was confused by my edit. – Makoto Kato Jul 26 '12 at 19:25

2 Answers2

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Not necessarily: if $\ell = 3$ then there are only six units in $A$ and none of them are totally imaginary.

David Loeffler
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The proposition is true:

Let $K^+$ denote the totally real subfield of $K$. Let $U$ be the group of units in $K$, and $U^+$ be the group of units in $K^+$. Dirichlet's Theorem implies that $U^+$ has finite index in $U$. I claim that in fact the index is odd. Indeed, suppose that $u$ is an element of whose image in $U/U^+$ is of exact order $2$.

Then $K = K^+(u) = K^+(\sqrt{u^2})$ is obtained by extracting the square root of a unit, and so is unramified over $K^+$ except possibly at primes lying over $2$. However, we know that $K/K^+$ is ramified at precisely the prime lying over $l$.

Consequently $U/U^+$ has odd order.

If $u \in U$ were purely imaginary, then $u \not\in U^+$, but $u^2 = - u \overline{u} = - |u|^2$ is an element of $U^+$, contradicting what we have just proved. Thus $U$ contains no purely imaginary elements.

[Hopefully this is correct; the previous argument I posted was nonsense.]

Matt E
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