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If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that αn=1 for some n). For example, (3/5 + 4/5 i) is not a root of unity.

But if we assume that α is an algebraic integer with absolute value 1, does it follow that α is a root of unity?


I know that if all conjugates of α have absolute value 1, then α is a root of unity by the argument below:

The minimal polynomial of α over Z is $\prod_{i=1}^d (x-\alpha_i)$, where the $\alpha_i$ are just the conjugates of α. Then $\prod_{i=1}^d (x-\alpha_i^n)$ is a polynomial over Z with αn as a root. It also has degree d, and all roots have absolute value 1. But there can only be finitely many such polynomials (since the coefficients are integers with bounded size), so we get that αn=σ(α) for some Galois conjugation σ. If σm(α)=α, then αnm=α.

Thus αnm-1=1.

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Consider (3+4i)/5. –  Pierre-Yves Gaillard Sep 9 '10 at 7:10
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@Pierre-Yves Gaillard: but that is not an algebraic integer. –  Qiaochu Yuan Sep 9 '10 at 7:36
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@Qiaochu Yuan : You're right! Thanks! Shame on me! Moreover, this is explicitly mentioned in the question. I have no excuse! [I often forget the very first rule: read carefully the question!] –  Pierre-Yves Gaillard Sep 9 '10 at 7:54
    
@Jonas Kibelbek : It seems to me you can also phrase your very nice argument as follows. Let $K\subset\mathbb C$ be Galois of finite degree over $\mathbb Q$. Then the algebraic integers of $K$ which lie, together with their conjugates, on the unit circle form a finite (multiplicative) group. –  Pierre-Yves Gaillard Sep 9 '10 at 11:09
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@Jonas Kibelbek - Here is a formulation of the argument by Kevin Buzzard: mathoverflow.net/questions/10911/… - This link is supposed to lead to KB's answer, but seems to lead to the question itself... At the time of writing, KB's answer is the first one. –  Pierre-Yves Gaillard Sep 17 '10 at 4:35
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4 Answers

up vote 27 down vote accepted

No. There are algebraic integers on the unit circle which aren't roots of unity.

This paper by Ryan Daileda provides some useful information and references.

Also see Salem numbers.

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Thank you! That's an excellent reference and paper (with the added benefit of being very clear and just 3 pages long)! I'm curious now what is the smallest extension containing such a unit. –  Jonas Kibelbek Sep 9 '10 at 8:13
    
What do you mean by smallest extension? Extension of least degree? Least discriminant (in abs. value)? –  KCd Sep 9 '10 at 12:01
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Let $x$ be an algebraic number with absolute value $1$. Then $x$ and its complex conjugate $\overline{x} = 1/x$ have the same minimal polynomial. Writing $f(T)$ for the minimal polynomial of $x$ over $\mathbb{Q}$, with degree $n$, the polynomials $T^nf(1/T)$ and $f(T)$ are irreducible over $\mathbb{Q}$ with root $\overline{x}$, so the polynomials are equal up to a scaling factor: $$T^nf(1/T) = cf(T).$$ Setting $T = 1$, $f(1) = cf(1)$.

Assuming $x$ is not rational (i.e., $x$ is not $1$ or $-1$), $f$ has degree greater than $1$, so $f(1)$ is nonzero and thus $c = 1$. Therefore $$T^nf(1/T) = f(T),$$ so $f(T)$ has symmetric coefficients. In particular, its constant term is $1$. Moreover, the roots of $f(T)$ come in reciprocal pairs (since $1$ and $-1$ are not roots), so $n$ is even.

Partial conclusion: an algebraic number other than $1$ or $-1$ which has absolute value $1$ has even degree over $\mathbb{Q}$ and its minimal polynomial has constant term $1$. In particular, if $x$ is an algebraic integer then it must be a unit.

There are no examples of algebraic integers with degree $2$ and absolute value $1$ that are not roots of unity, since a real quadratic field has no elements on the unit circle besides $1$ and $-1$ and the units in an imaginary quadratic field are all roots of unity (and actually are only $1$ and $-1$ except for $\mathbb{Q}(i)$ and $\mathbb{Q}(\omega)$). Thus the smallest degree $x$ could have over $\mathbb{Q}$ is $4$ and there are examples with degree $4$: the polynomial $$x^4 - 2x^3 - 2x + 1$$ has two roots on the unit circle and two real roots (one between $0$ and $1$ and the other greater than $1$).

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Thank you; I appreciate the thorough explanation. In my comment above, I was simply wondering about the extension of least degree, curious if degree 4 was enough-- as you've shown it is. Checking various symmetric polynomials in Wolfram Alpha, I see there are many such unimodular units. x^4-3x^3+3x^2-3x+1 has discriminant just -275. –  Jonas Kibelbek Sep 10 '10 at 2:09
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The answer to your question is no - not all algebraic integers with absolute value 1 are roots of unity - as you may have learned by now from the 2005 paper of Daileda that was referenced in the answer supplied by Robin Chapman. However, you should be aware that Daileda has in fact rediscovered these simple folklore results on unimodular units. These results are probably at least a half-century old - if not much older. Indeed, I recall reading similar results in papers published by Iwasawa in the fifties or sixties. I don't have the time now to locate said Iwasawa reference, but here is another reference that suffices to prove my point. This 1979 paper of Nakahata includes the same results and the same simple proofs found in Daileda's paper. Moreover, Nakahata places these results naturally into a more general context. Thus I highly recommend that you consult Nakahata's paper in addition to Daileda's. I've appended its Zbl review below.

alt text

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Thank you, it's nice to have another reference and to see a more general result. –  Jonas Kibelbek Sep 17 '10 at 3:35
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Let me first mention an example in Character Theory. Let $G$ be a finite group of order $n$ and assume $\rho$ is a representation with character $\chi:=\chi_\rho$ which is defined by $\chi(g)=Tr(\rho(g))$. Since $G$ is a finite group then, by invoking facts from linear algebra, one can show $\chi(g)\in\mathbb{Z}[\zeta_n]$. For abelian groups, it is easy to see $\chi(g)$ is a root of unity, when $\chi$ is irreducible, but what about non-abelian groups? In other words let $|\chi(g)|=1$, what can we say about $\chi(g)$?

This relates to your question. Let assume $K/\mathbb{Q}$ be an abelian Galois extension inside $\mathbb{C}$, and take an algebraic integer $\alpha\in\mathcal{O}_K$ such that $|\alpha|=1$, then for any $\sigma\in Gal(K/\mathbb{Q})$ we have $$ |\sigma(\alpha)|^2=\sigma(\alpha)\overline{\sigma(\alpha)} $$ Since $K/\mathbb{Q}$ is abelian then $\overline{\sigma(\alpha)}=\sigma(\overline{\alpha})$ so $$ |\sigma(\alpha)|^2=\sigma(|\alpha|)=1 $$ Then norm of all its conjugate is one so it must be a root of unity. This answer to the question was posed, therefore if $|\chi(g)|=1$ then $\chi(g)$ is root of unity.

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