# Degree of minimal polynomial over $\mathbb{Q}$

Prove that degree of minimal polynomial over $$\mathbb{Q}$$ of $$\zeta_{7}$$ , a primitive 7th root of unity is not a prime number.

I thought as $$\zeta_{7}$$ =$$1^{1/7}$$ so I can write $$1^{1/7}$$ =x which implies $$x^{7}$$ =1 . So, it's prime.

But answer is not prime. So, I am missing some concept.

Can anyone please tell what mistake I am doing.

The mistake is, $$x^7-1$$ is not the minimal polynomial of $$\zeta_7$$, as it is not irreducible over $$\Bbb{Q}$$. The minimal polynomial of $$\zeta_7$$ is the irreducible factor $$x^6+x^5+x^4+x^3+x^2+x+1$$ of $$x^7-1$$ over $$\Bbb{Q}$$.
• how is $x^{6}$ + ....+x +1 is minimal polynomial , you need to prove it!! – Ben Jul 1 at 17:43
• Yes. Recall the definition of minimal polynomial $\alpha$ over $F$. It is the irreducible monic polynomial over $F$ with $\alpha$ as a root. Now that polynomial is a cyclotomic polynomial which is irreducible over $\Bbb{Q}$ (can be shown from Eisenstein's criteria). Now $\zeta_7$ is a root of $x^7-1=(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)$ but not a root of $x-1$, so it must be root of the other factor. – user598858 Jul 1 at 17:50
• Eisenstein crieria can't prove $x^{6}$ + $x^{5}$ +... +1 to be irreducible . So, how can I prove $x^{6}$ + $x^{5}$ +... +1 to be irreducible. – Ben Jul 1 at 18:05
• Although I am giving a rough idea of the proof. Let denote that polynomial by $p(x)=\frac{x^7-1}{x-1}$ then you can show $p(x+1)$ is irreducible polynomial over $\Bbb{Q}$ by Eisenstein's criteria using the prime 7. Now $p(x+1)$ is irreducible implies $p(x)$ is so. – user598858 Jul 1 at 18:13
• Well, you try to prove if $p(x)$ is reducible then for any $a$, $p(x+a)$ is also reducible. It is easy. – user598858 Jul 1 at 18:33
A minimal polynomial should be irreducible. But all its roots are non-real roots of 1. Since coefficients are real (even rational) all roots of the minimal polynomial must be decomosed into pairs $$(a, \bar a)$$ , so the number of roots and the degree must be even. The only even prime is 2 but a quadratic rational polynomial cannot have a non-real root which is the seventh root of 1. So, indeed, the degree of minimal polynomial is not prime.