How many primitive polynomials are there in $\mathbb{F}_p[x]$ where $p$ is prime?

Question

I've read the post The number of primitive polynomials of degree $m$ over a finite field $GF(p^m)$, but I don't understand why there can't be common roots of distinct primitive polynomials of the same degree. Any help would be highly appreciated! Thank you in advance!

Answer 1

• Primitive polynomials are irreducible
• If two polynomials $f, g$ have roots in common, then $\gcd(f, g)\neq 1$

Couple these two facts together, and you get that two primitive polynomials with any roots in common must have all their roots in common.

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Somewhat alternate reasoning (and the one hinted at in the linked post):

Given a primitive polynomial $f$ over $\Bbb F_p$, extending $\Bbb F_p$ by any of the roots of $f$ gives $\Bbb F_{p^m}$. Given another primitive polynomial $g$ of the same degree, any one of its roots generates the same (or at least an isomorphic) field.

If $f$ and $g$ have roots in common, then $h = \gcd(f, g)$ will also have those roots in common. And thus any root of $h$ will also generate $\Bbb F_{p^m}$. But that's a degree $m$ extension, so $h$ must have degree $m$. Thus $\gcd(f, g)$ has the same degree as $f$ and $g$, so $f$ and $g$ are (up to multiplication by a constant) the same polynomial.