Question

Let p be prime and F (x) an irreducible polynomial of degree 2 in ${\mathit{Z}}_{\mathbf{p}}\left[x\right]$. If K is an extension field of ${\mathbf{Z}}_{\mathbf{P}}$of degree ${\mathbf{p}}^{\mathbf{3}}$prove that F (x) is irreducible in K [x].

Short answer

It is proved that F [x] is irreducible in K [x].

Step-by-step solution

Given

Let p be prime and F (x) is irreducible polynomial of degree 2 in $Z_P\left[x\right]$. If K is an extension field of $Z_P\left[x\right]$ with order $p^3$then to prove that $F\left(x\right)$is irreducible in K [x].

Showing that f (x) is irreducible in K [x]

Let on the contrary F (x)is irreducible in K (x). As K is an extension field of $Z_P\left[x\right]$with order $p^3$and K is the splititing field of the polynomial $f\left(x\right)=x^{p^n}-x$over $Z_P$. But f (x) is an irreducible polynomial of degree 2 in $Z_P\left[x\right]$so it must be of the form:

$f\left(x\right)=x^2+1,x^3+x+2,....$

And observe that any of these polynomial cannot be written in the form $f\left(x\right)=x^{p^n}-x$over $Z_P$. Therefore it contradicts the assumption that K is the splititing field of the polynomial $f\left(x\right)=x^{p^n}-x$over $Z_P$and f (x) is reducible in K [x] .

Thus is irreducible polynomial in .