$f(x)\in\mathbb{Z}[x]$, $\frac{r}{s}\in\mathbb{Q}$ is a root of $f(x)$ with $(r,s)=1$. Show that $r-ms$ is a divisor of $f(m)$ for any integer $m$
106401
Prove: $p-mq | f(m)$ where $m$ is any integer
102553
How to show that $f(z) = f(i)$
2907
How do I show that $x$ is a root of $p(x)$ in $F[x]/\langle p(x)\rangle$?
9129
Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$
158182
Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.
116585
$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
17926
Zero divisor in $R[x]$
131692
How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
96087
Let $\mathbb{F}_2 \cong \mathbb{Z}/2\mathbb{Z}$. Is $x^4+x^2+1$ irreducible in $\mathbb{F}_2[x]$?
130944
Greatest common divisor of polynomials over $\mathbb{Q}$
119305
Is the splitting field equal to the quotient $k[x]/(f(x))$ for finite fields?
145722
Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
138026
To show that $n!f(x) \in \mathbb{Z}[x]$
31555
Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
23071
Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$
116970
On the number divisors of a number
6722
to show $f$ is a polynomial in $z$