Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic.

Question

Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic.

I've proven that $X^4 + X + 1$ is irreducible, so $L$ is a field. I also know that $X^5 + X + 1$ is not irreducible. Also I've proven that $|L| = 16$.

Could someone help me out proving that $L^*$ is cyclic ?

Answer 1

since the group is of order 15 any element must have order 1,3,5 or 15.

in any field the equation $x^k = 1 $ can have at most k roots, so the number of elements with order less than $15$ is at most $1 + 3 + 5 = 9$

hence there must be an element of order 15

i will add, after the query from OP, that a slightly more sophisticated form of this argument will show that for any finite field its multiplicative group is cyclic.

Answer 2

The multiplicative group of a finite field is always cyclic.

Answer 3

Calculate $X,X^2,\ldots, X^{15}$ (modulo $X^4+X+1$) and show that you get everything in $L^*$.