# Why algebraic integer?

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?

(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 $$|x|.
Proof: We have $$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$$.