show that $pO_K=(p,(1-\zeta_p)^{p-1}.$
118305
Let $p$ be an odd prime. $p$ and $(1-\zeta_p)^{p-1}$ are associates in $\mathbb{Z}[\zeta_p]$.
131881
Show that the roots of P' lie in the same half plane as the roots of P
96062
Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
18634
Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
158106
Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
151805
Prove that $\tau (x_1 x_2 ... x_k) \tau^{-1} = (\tau(x_1) \tau(x_2) ... \tau (x_k))$
121661
Evaluating the sum $\Sigma \, \frac{1}{k(k+p)}$
127169
Show that the zeros of $\sum_{k=1}^n kz^{k-1}$ are inside the unit disc
144819
Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
102234
Show $a^p \equiv a \mod 2p$
46821
Finding roots in $\mathbb{Z}_p$
22524
Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$
66877
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
18108
Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups