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I face problems whenever I am asked to work with fields of positive characteristic and roots of unity over those fields.I am stuck with a similar problem and looking for some help regarding this problem.The problem is as follows:

If $\omega_1$ and $\omega_2$ are primitive $n$ th roots of unity over a finite field $k$ of characteristic $p>0$ and $n\neq p$.Then is it true that $\omega_1$ and $\omega_2$ have the same minimal polynomial over $k$.

I want to know exactly why we need the field to be finite field.Can someone give me a detailed answer or explanation?

Kishalay Sarkar
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1 Answers1

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The answer to the question is "No!!", in general (follow the link below for a more precise answer).

Possibly the simplest conterexample is in the field $\Bbb{F}_5$, where both $2$ and $2^{-1}=3$ are primitive fourth roots of unity with minimal polynomials $x-2$ and $x-3$ respectively.

If you think that was a cheating example, and (like me) prefer characteristic two examples then the simplest examples reside in the field $\Bbb{F}_8$. Modulo two we have the polynomial factorization $$ x^7-1=(x-1)(x^3+x+1)(x^3+x^2+1). $$ Implying that the roots of both cubic factors are seventh roots of unity. Seven being a prime they are all automatically primitive.

Over $\Bbb{Q}$ the minimal polynomial of the roots of unity of order $n$ is the cyclotomic polynomial $\Phi_n(x)$, and the game to be played is how that factors modulo $p$. There are no repeated factors when $\gcd(p,n)=1$, and the conclusion is that the primitive roots of unity share the same minimal polynomial over $\Bbb{F}_p$ if and only if $\Phi_n(x)$ remains irreducible after reduction modulo $p$.

But the cyclotomic polynomial is rarely irreducible over the prime field. I have touched its factorization under various guises more often than I care to admit. Start for example here and check out the list of linked questions.

However, all the primitive roots of a given order (in some extension of $\Bbb{F}_p$) do have minimal polynomials of the same degree. This is because a field extension generated by one primitive root necessarily contains all the others (in a sense that can be made more precise for example by requiring that we work inside a fixed algebraic closure). Hence the minimal polynomials share the degree with that of the said field extension.

Jyrki Lahtonen
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