Prove that a monic sum is irreducible over a field.

Question

1. Prove that $\sum_{k = 0}^{10} X^k$ is irreducible over $\mathbb{F}_2$.
2. Prove that $\sum_{k = 0}^{20} X^k$ is the product of all irreducible polynomials of degree $2$ or $3$ and two polynomials of degree $6$ over $\mathbb{F}_2$.

I have no idea how to prove either...

Answer 1

Hints.

(1) First note that $1+x+\cdots+x^{10}$ divides $x^{2^{10}}-x$. Thus $1+x+\cdots+x^{10}$ must have as its prime factors uniqud primes of degree $1,2,5,$ or $10$.

Then show that $x^{2^5}-x$ and $x^{2^2}-x$ are relatively prime to $1+x+\cdots+x^{10}$ to get that it must only have prime factors of degree $10$ - that is, it must be prime.

(2) Use that $$1+x+\cdots + x^{20} = \frac{x^{2^6}-x}{x(1-x)(1+x^{21}+x^{42})}$$