On a certain property of units in the cyclotomic number field of an odd prime order

Question

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 $\epsilon$ be a unit of $A$.

My question: Is $\epsilon/\bar{\epsilon}$ a root of unity?

Motivation and Effort This is clear from this question.

What's the reason for the downvotes? Unless you tell me, I can't improve my question. — Makoto Kato, Jul 27 '12 at 07:14
(I edited your comment into what I think you meant to write. Beware of double negatives!) — Willie Wong, Jul 27 '12 at 09:02

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

2

Yes. In the notation of this answer, the ratio $\epsilon/\overline{\epsilon}$ lies in $U^-$, which is a finite group (as explained in the linked answer).

edited Apr 13 '17 at 12:20

Community

1

answered Jul 26 '12 at 03:49

Matt E

123,735

Conjecture · Answer 2 · 2019-09-13T00:51:21.683

0

Yes: By Kummer's Lemma, $\epsilon/\bar{\epsilon}$ is a unit in A as it is a power of a $l$-th root of unity

edited Sep 13 '19 at 00:51

answered Sep 13 '19 at 00:45

Conjecture

3,088

On a certain property of units in the cyclotomic number field of an odd prime order

2 Answers2

Linked