query: If $K$ is a finite field, then $|K|=p^d$ where $p$ is prime and $d\geq 1$.
positive_passages.docid: 3703
positive_passages.text: Order of finite fields is $p^n$
negative_passages.docid: 99205
negative_passages.text: Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
negative_passages.docid: 142530
negative_passages.text: Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.
negative_passages.docid: 7899
negative_passages.text: Show that if $p$ is prime then $\Bbb Z_p$ is a field
negative_passages.docid: 68145
negative_passages.text: If $p$ is an odd prime and $k$ an integer with $0<k<p-1$ then $1^k + 2^k + \ldots + (p-1)^k$ is divisible by $p$
negative_passages.docid: 108380
negative_passages.text: Assume $P$ is a prime ideal s.t. $K \subset P$, show $f(P)$ is a prime ideal
negative_passages.docid: 134963
negative_passages.text: Proving a non-prime (composite) number $> 1$ has a divisor $d \leq\sqrt q$
negative_passages.docid: 151985
negative_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
negative_passages.docid: 152109
negative_passages.text: In a UFD if $x$ is irreducible, then $(x)$ is prime
negative_passages.docid: 62239
negative_passages.text: For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
negative_passages.docid: 63659
negative_passages.text: If D is an Integral Domain and has finite characteristic p, prove p is prime.
negative_passages.docid: 23153
negative_passages.text: $\bmod p\!:\ x^2\equiv 1\iff x\equiv \pm1\ $ for prime $p$
negative_passages.docid: 97196
negative_passages.text: $k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$
negative_passages.docid: 31724
negative_passages.text: How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$?
negative_passages.docid: 10403
negative_passages.text: Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
negative_passages.docid: 130466
negative_passages.text: Proving every prime ideal is maximal in a finite dimensional ring over a field.
negative_passages.docid: 51465
negative_passages.text: New, elegant proofs for $\varphi(p^{k})=p^{k}-p^{k-1}$