English reference for a result of Kronecker?

Question

Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference:

Let $f$ be a monic polynomial with integer coefficients in $x$. If all roots of $f$ have absolute value at most 1, then $f$ is a product of cyclotomic polynomials and/or a power of $x$ (that is, all nonzero roots are roots of unity).

However, I don't have access to this article, and even if I did my 19th century German skills are lacking; does anyone know a reference in English I could check for details of the proof?

Answer 1

I don't know a reference, but here is a quick proof: Let the roots of the polynomial be $\alpha_1$, $\alpha_2$, ..., $\alpha_r$. Let

$$f_n(x) = \prod_{i=1}^r (x- \alpha_i^n).$$

All the coefficients of $f_n$ are rational, because they are symmetric functions of the $\alpha$'s, and are algebraic integers, because the $\alpha$'s are, so they are integers. Also, since $|\alpha_i| \leq 1$, the coefficient of $x^k$ in $f_n$ is at most $\binom{r}{k}$.

Combining the above observations, the coefficients of the $f_n$ are integers in a range which is bounded independent of $n$. So, in the infinite sequence $f_i$, only finitely many polynomials occur. In particular, there is some $k$ and $\ell$, with $\ell>0$, such that $f_{2^k} = f_{2^{k + \ell}}$. So raising to the $2^{\ell}$ power permutes the list $(\alpha_1^{2^{k}}, \ldots, \alpha_r^{2^k})$. For some positive $m$, raising to the $2^{\ell}$ power $m$ times will be the trivial permutation. In other words,

$$\alpha_i^{2^k} = \alpha_i^{2^{k+\ell m}}$$.

Every root of the above equation is $0$ or a root of unity.

Answer 2

If all the Galois conjugates of an algebraic integer $\alpha$ have absolute value at most 1, then the norm of this algebraic integer is a rational integer with absolute value at most 1. Hence either the algebraic integer is 0, or its norm is $\pm1$, and in the latter case all the Galois conjugates of $\alpha$ must have absolute value equal to 1. Now it's a well-known fact that the only algebraic integers all of whose conjugates have absolute value 1 are the roots of unity [Proof: bounds on the absolute values of the conjugates give bounds on the coefficients of the min polys, and so there are only finitely many possible min polys for $\alpha^n$, $n=1,2,3,\ldots$ (as the degrees are bounded too), and hence $\alpha^n=\alpha^m$ for some $m>n>0$], so there is a complete proof for you.

Answer 3

Another nice reference (with a short proof) is

G. Greiter, A simple proof for a theorem of Kronecker, Amer. Math. Monthly 85 (1978), no. 9, 756–757.

The proof in this paper is related to the proofs given above by Kevin and David, but is a bit more elementary.

Answer 4

Bombieri and Gluber's recent book "Heights in Diophantine Geometry" has a proof of this in chapter 1.

Answer 5

Here is a good reference by P.A. Damianou!

http://www.mas.ucy.ac.cy/~damianou/kronecker.pdf

Answer 6

In case anyone is interested, here's a run-down of the history of this result and a comparison of available proofs, as far as I could uncover in a rainy evening.

• To be clear, Kronecker's original article, cited in P.A. Damianou's article, actually proves the following: given a monic polynomial with integer coefficients all of whose roots have norm 1, the roots are all roots of unity.
• Kronecker's argument (in German) is almost identical to David Speyer's above. Both are almost identical to P.A. Damianou's---Damianou seems to have largely translated Kronecker, I think.
• G. Gretier's Monthly article (cited by Damianou), J. Spencer's Fibonacci Quarterly article (cited by Greiter), and Kevin Buzzard's argument in brackets all prove this statement in related but different ways. Greiter uses companion matrices and diagonalization. Spencer's argument is the longest and builds a Fibonacci-type linear recurrence out of such a polynomial, using the corresponding Binet-type formula to show the recurrence must repeat.
• Bombieri and Gluber's version is essentially a repackaging of Kronecker's argument in more number-theoretic language.
• Polya and Szego's question 200 seems to be mostly just the theorem statement, though questions 198 and 199 are similar to the first part of Kevin Buzzard's argument.

Answer 7

When I was a kid the standard reference for this result was Polya and Szego, Problems and Theorems in Analysis, Volume 2. It's question 200 in Part 8.