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I am going through a past MIT 2005 paper and found this tricky question, which is baffling me. Can anybody help? Let $a_1,a_2,\ldots,a_n$ be algebraic numbers such that $a_1^i + a_2^i + \ldots a_n^i \in \mathbb{Z}$ for all positive integers $i$. Prove that $a_1,a_2,\ldots, a_n$ are algebraic integers. (Hint: what is the radius of convergence of the formal power series of $\sum_{j=1}^n 1/(1-a_jt)$ over $\mathbb{Q}_p$, ie p-adic numbers.

Working 1: The power series is $1 + (a_1+\ldots + a_n)t + (a_1^2 + \ldots + a_n^2)t^2 + \ldots$ so it is an infinite polynomial with integer coefficients with radius of convergence $min \{1/|a_1|,\ldots,1/|a_n|\}$.

Working 2: I think Hensel's lemma might help, but not sure how.

Anybody with ideas on how to proceed on this interesting and tricky question?

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1 Answer 1

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(This is probably obvious but I will still make this obligatory remark since it wasn't mentioned before.) Let $K=\Bbb Q(a_1,\dots,a_n)$. Since $\mathcal O_K=\bigcap_{\mathfrak p} \mathcal O_{K,\mathfrak p}$ it suffices to prove that $a_1,\dots,a_n$ have non-negative valuation at all primes $\mathfrak p$. Thus we look at all possible embeddings $K\to\Bbb C_p$ for all primes $p$ and look at $a_1,\dots,a_n$ in $\Bbb C_p$ (or $K_\mathfrak p$ instead of $\Bbb C_p$).

Lemma: Let $f(T)=\sum_{i=0}^\infty a_iT^i$ be a power series in $\Bbb C_p[[T]]$ (or over any valued field). Suppose that formally $f(T)=g(T)/h(T)$ where $g(T),h(T)\in\Bbb C_p[T]$ are coprime polynomials. If $f(T)$ is convergent with radius of convergence $R$, then $g(T)/h(T)$ is regular at $x$ and $f(x)=g(x)/h(x)$ for all $x$ with $|x|<R$.
Proof: We have $h(T)f(T)=g(T)$. It is a general result that evaluation and formal product of convergent power series commute (see e.g. Robert 'A Course in $p$-adic Analysis' Chapter 6, Section 1.2, Proposition 2), hence $h(x)f(x)=g(x)$ and it is not possible that $h(x)=0$ since $h$ and $g$ are coprime.

Now let $f(T)=n+(a_1+\dots+a_n)T+(a_1^2+\dots+a_n^2)T^2+\dots$. As formal series we have $f(T)=\sum_{i=1}^n \frac{1}{1-a_iT}=\frac{g(T)}{h(T)}$ where $g(T)=\sum_{i=1}^n\prod_{j\ne i}(1-a_jT)$, $h(T)=(1-a_1T)\dots(1-a_nT)$. Now, by assumption $f(T)\in\Bbb Z_p[[T]]$, thus $f$ has radius of convergence at least $1$, i.e. converges in $U=\{x\in\Bbb C_p\mid |x|<1\}$. By the lemma the rational function $g(T)/h(T)$ is regular in $U$. But notice that $a_i^{-1}$ will always be a pole of $g(T)/h(T)$ (if the $a_i$ are not distinct, $g(T)$ and $h(T)$ are not coprime but $(1-a_iT)$ always appears one more often in $h(T)$ than in $g(T)$), hence $a_i^{-1}\notin U$, i.e. $|a_i|\leq1$.

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  • $\begingroup$ thanks for your comprehensive answer. it is opening the skies for me now. $\endgroup$
    – ringguy
    Commented Sep 23, 2021 at 3:01

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