Let $p$ be prime and $n = p^2$. Show that the polynomial $1+px$ has a multiplicative inverse in $\mathbb{Z}_n[x]$.
130104
Show $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$ (inverting unit + nilpotent)
145112
Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic
149544
maximum of monic polynomial on unit circle is 1 implies $p(z)=z^n$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
129834
Show that there is a unique polynomial of degree at most $2n+1$ such that $q^{[k]}(x_1)=a_k,$ $q^{[k]}(x_2)=b_k$ for $k=0, \dots, n$.
124131
Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$
131289
Polynomial with a root modulo every prime but not in $\mathbb{Q}$.
344
Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
154945
$|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that $|rx^2+qx+p|\le2$ for all $x$ in $[-1; 1]$
90144
Show that if $a+bi$ is irreducible over $\mathbb{Z} [i]$ then $a^2 + b^2$ is prime over $\mathbb{Z} $
43940
Showing that $n$ is pseudoprime to the base $a$
81063
If $p$ is a polynomial, then either $\,p(z)=z^n\,$ or $\,\max_{|z|=1}|p(z)|>1$.
138026
To show that $n!f(x) \in \mathbb{Z}[x]$
6722
to show $f$ is a polynomial in $z$
11693
If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
121714
Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible