Prove that $t^4 + t^3 + t^2 + t+1$ is not a primitive polynomial for $\mathbb{F}_2$

Question

I tried showing that $f(t) = t^4 + t^3 + t^2 + t+1$ is reducible over $\mathbb{F}_2$, but it turns out that $f(t)$ is irreducible. So I have to check whether or not $t$ is a primitive element of $\mathbb{F}_2$, i.e. $t$ generates the cyclic group $\mathbb{F}_1 ^* = \{ 1 \}$. However $t^1 \ne 1$ and therefore $f(t)$ is not primitive.

Is this correct? Thanks.

Answer 1

In field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite extension field GF$(p^m)$.

Definition, straight from Wikipedia. The root doesn't have to be in the base field. It shouldn't be in the base field.

Is the root going to be a primitive element in the larger field's multiplicative group? Well - this is a cyclotomic polynomial. Since $(t-1)f(t)=t^5-1$, any $t$ that is a root is a fifth root of unity. Since $1$ isn't a root, that means it's a primitive fifth root of unity. The only way that can generate the multiplicative group for a field is if that field has exactly five invertible elements - so, then, it has exactly six total elements. There are no such fields. No matter what prime we choose, this isn't a primitive polynomial.

Answer 2

As $f$ is irreducible, $f$ is primitive iff any of its zeros, $\alpha$ say, generates the cyclic group $\Bbb F_{2^4}^*$, which has order $15$. But $f(t)\mid (t^5-1)$ as polynomials, so $\alpha^5=1$.