The document has theorems and problems about polynomials with integer coefficients.

For a simple example, if some integer value has few factorizations (e.g. a unit $\,\pm1 $ or prime $p$) then the polynomial must also have few factors, asssuming that that the factors are distinct at the …

Definition. An integer polynomial P (x) is a polynomial of the form cnxn + cn−1xn−1 + . . . + c1x + c0 where cn, cn−1, . . . , c0 ∈ Z. Integer polynomials problems span across many ideas in both algebra …

Let p(x) be a polynomial with integer coeʀ cients. Suppose that for some positive integer c, none of p(1), p(2), . . . , p(c) are divisible by c. Prove that p(b) is not zero for any integer b.

Jul 9, 2025 · To find the simplest polynomial function with integer coefficients having the given zeros, we need to consider all the provided roots: 4i,3,−1. Since complex roots must come in conjugate pairs, …

Sep 30, 2011 · We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers.

Suppose that P(x) = f(x)g(x), where f and g are integer polynomials. Since P has only one zero with modulus not less than 1, one of the polynomials f, g, has all its zeros strictly inside the unit circle.