query: $f(x)$ is monoic polynomial , prove that if $f(k), f(k + 1),..., f(k + p)$ is not divisible by $p + 1$, then $f(x) = 0$ has no rational solution.
positive_passages.docid: 155642
positive_passages.text: Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring
negative_passages.docid: 65619
negative_passages.text: Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?
negative_passages.docid: 43511
negative_passages.text: Proving products of coefficients of two polynomials are zero when the product of the polynomials is zero
negative_passages.docid: 30511
negative_passages.text: Monic $f(x)\in\Bbb Z[x]$ has no rational root if $f(0)\ \&\ f(1)$ odd [Parity Root Test, Modular Root Test]
negative_passages.docid: 92537
negative_passages.text: Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots
negative_passages.docid: 38805
negative_passages.text: Prove that $x^{2} \equiv -1 \pmod p$ has no solutions if prime $p \equiv 3 \pmod 4$.
negative_passages.docid: 144222
negative_passages.text: If $f(x + 1) + f(x − 1) = f(x), \forall x \in \mathbb{R}$,then how to find $k$ such that $f(x + k) = f(x)$?
negative_passages.docid: 42265
negative_passages.text: A proof that $\displaystyle\sum_{i=1}^{n}P''(r_i)/P'(r_i)=0$ for a degree $n$ polynomial
negative_passages.docid: 33099
negative_passages.text: If $f$ is an entire function where every power series expansion has at least one 0 term, show it is a polynomial
negative_passages.docid: 116856
negative_passages.text: Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 117816
negative_passages.text: Is $\frac{x^2+x}{x+1}$ a polynomial?
negative_passages.docid: 65901
negative_passages.text: Solution verification to complex analysis problem, prove $f$ is a polynomial.
negative_passages.docid: 121076
negative_passages.text: Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
negative_passages.docid: 56171
negative_passages.text: Is $K(X)$ never algebraically closed?
negative_passages.docid: 81921
negative_passages.text: If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
negative_passages.docid: 40014
negative_passages.text: A polynomial that is zero on an open set
negative_passages.docid: 90533
negative_passages.text: Show that if $p(x)$ is reducible in $F[x]$, then $F[x]/(p(x))$ is not an integral domain.