query: If $p$ a prime number $\mathbb{Z}^{\times}_p$ is always isomorphic to $(\mathbb{Z}_{p-1},+)$
positive_passages.docid: 145112
positive_passages.text: Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic
negative_passages.docid: 154754
negative_passages.text: Is $O_n$ isomorphic to $SO_n \times \{\pm I\}$?
negative_passages.docid: 141418
negative_passages.text: When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?
negative_passages.docid: 121890
negative_passages.text: Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
negative_passages.docid: 78118
negative_passages.text: Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
negative_passages.docid: 44033
negative_passages.text: Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
negative_passages.docid: 118305
negative_passages.text: Let $p$ be an odd prime. $p$ and $(1-\zeta_p)^{p-1}$ are associates in $\mathbb{Z}[\zeta_p]$.
negative_passages.docid: 44083
negative_passages.text: prove that maximal ideal in $\mathbb{Z}$ generated by a prime number
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 100426
negative_passages.text: Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
negative_passages.docid: 5142
negative_passages.text: Infinitely many primes of the form $pn+1$
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 70628
negative_passages.text: Prove p-adic valuation is completely additive for prime p
negative_passages.docid: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 138058
negative_passages.text: Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
negative_passages.docid: 120183
negative_passages.text: Prove that $S_3\times\mathbb{Z}_2$ is isomorphic to $D_6$
negative_passages.docid: 81238
negative_passages.text: Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?