query: $p$ a prime and $A=\{\frac{a}{b}\in\mathbb{Q}:p\text{ does not divide } b\}$ is a PID and decribe its units
positive_passages.docid: 24931
positive_passages.text: Is the localization of a PID a PID?
negative_passages.docid: 42932
negative_passages.text: Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.
negative_passages.docid: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 53945
negative_passages.text: The square root of a prime number is irrational
negative_passages.docid: 126199
negative_passages.text: Proof of $\pi$ not being a quadratic irrational number.
negative_passages.docid: 85206
negative_passages.text: Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
negative_passages.docid: 155886
negative_passages.text: let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$
negative_passages.docid: 140948
negative_passages.text: Suppose $p$ and $q$ are odd primes and $p = q + 4a$ for some $a$. Prove that $(\frac ap) = (\frac aq)$ holds.
negative_passages.docid: 90055
negative_passages.text: Show that $\mathcal P(x)\notin x$
negative_passages.docid: 48587
negative_passages.text: Why is $1$ not a prime number?
negative_passages.docid: 136855
negative_passages.text: Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID
negative_passages.docid: 147777
negative_passages.text: If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?
negative_passages.docid: 54358
negative_passages.text: Why $(1-\zeta)$ unit where $\zeta$ is a primitive nth and n divisible by two primes
negative_passages.docid: 21747
negative_passages.text: A subring of the field of fractions of a PID is a PID as well.
negative_passages.docid: 153664
negative_passages.text: When is $\Bbb{Z[\zeta_n]}$ a PID?
negative_passages.docid: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 61899
negative_passages.text: Prove that the language $\{bin(p) \mid p\ \text{is prime}\}$ is not regular (prime numbers)