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The minimal polynomial of $\cos(2\pi/n)$ (by William Watkins and Joel Zeitlin, The American Mathematical Monthly Vol. 100, No. 5 (May, 1993), pp. 471-474) has full clarity on this matter (just take th …

There is a following result which is quite lovely, I think (I don't remember right away whose result this is): Let us define a function $f\colon\mathbb{N}\to\mathbb{Q}^+$ as follows: $f(1)=1$, and al …

If this were true, then you would prove that $\Phi_m(x)$ and $\Phi_n(x)$ are coprime after reduction modulo $p$, which is far from true. For instance, $\Phi_4(x)=x^2+1$ and $\Phi_2(x)=x+1$ are not cop …

We have $$\sum_k\frac{(-1)^kP(nk)}{k}=\sum_{k,p}\frac{(-1)^k}{kp^{nk}}=-\sum_p\ln\left(1+\frac{1}{p^n}\right)=\sum_p\ln\left(\frac{1-\frac{1}{p^{n}}}{1-\frac{1}{p^{2n}}}\right)=\ln\frac{\zeta(2n)}{\ze …

We have $$\sin\frac{\pi}{9}+\sin\frac{2\pi}9-\sin\frac{4\pi}9=\sin\frac{2\pi}{18}+\sin\frac{4\pi}{18}-\sin\frac{8\pi}{18}=\sin\frac{2\pi}{18}-\sin\frac{8\pi}{18}+\sin\frac{14\pi}{18},$$ and the latte …

I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote hi …

Let me try to summarize what I said in the comment to your question and what I implicitly used in what did not fit in the comment. As you will instantly see, the answer is lengthy because of high-scho …

One reference where the asymptotic result you are asking for was first established (I think), as well as some reasonable growth estimates for $a_n$, is D.G.Kendall and R.A.Rankin, "On the number of A …

Such a generalization (Dumas' theorem) was discussed here: Is a polynomial with 1 very large coefficient irreducible? A good source to learn about it is Prasolov's book on polynomials: http://tinyurl …

My personal experience suggests that reaching reasonable breadth of mathematical scope is achieved through three different mechanisms : 1) Attending talks (seminars, colloquia, workshops) in subjects …

1) For your third interpretation I have at least one relevant experience of my own - "factor systems" (some) number theorists deal with when talking about central simple algebras over number fields di …

One thing I completely forgot of was reminded to me by the reference to Feit-Thompson conjecture: applications of number theory/basic Galois theory to characters of finite groups and to structure theo …

Use the standard notations $e_k=\sum_{A\subset \{1,\dots,n\}, |A|=k} \prod_{i\in A} x_i$, with the conventions $e_0=1$ and $e_m=0$ for $m>n$; $p_k=\sum_{i=1}^n x_i^k$. If $n=p$, the statement is true …

A follow-up to the comment of Anonymous which addresses your question exactly: see slides 9-13 here for an investigation of your question. Basically, it can be proved that the decomposition into a pro …

What is true for sure is that this statement (in fact, a more general one) holds over a finite field (see Brawley, J. V., Carlitz, L. Irreducibles and the composed product for polynomials over a finit …