How many monic primitive quadratic polynomials are there in $Z_{7}[x]$?

Question

A theorem states that "for each prime p and for each integer $n \ge 1$, there exists a monic irreducible polynomial of degree n in $Z_{p}[x]$". I am not sure if this theorem will help answer my question, but can anyone explain please?

Answer 1

There exists $\varphi(p^n-1)$ primitive elements over $\mathbf F_p\,$ in $\,\mathbf F_{p^{\scriptstyle n}}$ and a given minimal polynomial of a primitive element has $n$ roots, which are also primitive elements. Hence the number of such minimal polynomials is $$\dfrac{\varphi(p^n-1)}n.$$ In the present case, this number is equal to: $$\frac{\varphi(48)}2 = \frac{\varphi(2^4)\varphi(3)}2=8.$$