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Abstract Algebra

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Gauss's Lemma on Primitive Polynomials with Integer Coefficients

ABSALG-EUVRK2

Let $P\in{\mathbb Z}[x]$ be a non-zero polynomial.

We say that $P$ is primitive when the greatest common positive divisor of the coefficients of $P$ equals $1$.

Let:

$P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0,\qquad Q(x)=b_mx^m+b_{m-1}x^{m-1}+\ldots+b_0$

be non-zero primitive polynomials in ${\mathbb Z}[x]$.

Let $p\ge 2$ be a prime number.

Which of the following is always true?​

A

For every $k\ge 0$ the prime $p$ does not divide $\sum_{i+j=k}a_ib_j$

B

There is a $k\ge 0$ such that $\sum_{i+j=\ell} a_ib_j$ is not divisible by $p$ for all $\ell\le k$

C

There is a $k\ge 0$ such that $\sum_{i+j=\ell} a_ib_j$ is not divisible by $p$ for all $\ell\ge k$

D

There is a $k\ge 0$ such that:

$\sum_{i+j=k} a_ib_j\equiv a_Ib_J\not\equiv 0$ mod $p$

for just one choice of $I$, $J$ with $I+J=k$

E

None of the above is always true