query: Show that $x^{p^n}-x$ is the product of all monic irreducible polynomials in $\mathbb{Z}/p\mathbb{Z}[x]$ of a degree $d$ dividing $n$.
positive_passages.docid: 8164
positive_passages.text: Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q}$?
negative_passages.docid: 109610
negative_passages.text: Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$
negative_passages.docid: 102432
negative_passages.text: Irreducible polynomial of degree $8$
negative_passages.docid: 46951
negative_passages.text: For what values of $n > 1$ and a is $x^n - a$ irreducible in $\mathbb Q[x]$?
negative_passages.docid: 137669
negative_passages.text: $P$ is a monic polynomial of degree $n$ , then which are correct?
negative_passages.docid: 80144
negative_passages.text: Prove that $x-1$ divides $x^n-1$
negative_passages.docid: 52528
negative_passages.text: Showing that minimal polynomial has the same irreducible factors as characteristic polynomial
negative_passages.docid: 25471
negative_passages.text: Irreducibility of $x^n-x-1$ over $\mathbb Q$
negative_passages.docid: 73344
negative_passages.text: Definition of irreducible polynomial via the units of an integral domain.
negative_passages.docid: 44878
negative_passages.text: Factor $x^5+x^2+1$ into irreducible polynomials in $Z[x]$
negative_passages.docid: 84697
negative_passages.text: Show that if $|f(z)| \leq M |z|^n$ then $f$ is a polynomial max degree n
negative_passages.docid: 24626
negative_passages.text: Irreducibility of $X^n - 2$ over $\mathbf Z[i]$.
negative_passages.docid: 97729
negative_passages.text: Show that a polynomial $f(x)$ over a field $k$ is irreducible if and only if the polynomial $f(x+1)$ is irreducible
negative_passages.docid: 77620
negative_passages.text: Irreducible polynomials over $\mathbb Q$ and $\mathbb Z $
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 27942
negative_passages.text: showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 157031
negative_passages.text: Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$