459 views

### Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
87 views

### How many irreducible factors of grade $6$ there is in $\mathbb{F}_{2}\left[ x\right]$? [duplicate]

How many irreducible factors of grade $6$ there is in the polynomial ring $\mathbb{F}_{2}\left[ x\right]$? I have solved this by using the fact that every irreducible polynomial of grad $i$ is a ...
28 views

### Is there a formula which would let me know how many irreducible polynomials there are to the power n, in $z_n$? [duplicate]

I found that $x^2+x+1$ is the only polynomial to the power 2 that is irreducible in $z_2$. Moreover I found that $x^3+x+1$ and $x^3+x^2+1$ are the only polynomials to the power 3 that are ...
22 views

### Number of irreducible polynomials of degree 4 on $\boldsymbol{Z}_3[x]$. [duplicate]

Find the number of irreducible polynomials of degree 4 in $\boldsymbol{Z}_3[x]$. Not really sure what to do here, I've tried listing them all out but this seems tedious, and all I need to find is the ...