Abstract Algebra: prove it is cyclic

Question

I have question in referring to below link.

Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic?

I know is f(x) is 16 elements. And vector space between M -> K is 2. Also we can know that vector space between $\mathbb{Z}_2$ -> K is 2. Therefore K has 4 element (which will be I suppose (x$^3$,x$^2$,x,1)). Now how can I know that it is cyclic.

Note: K is non-zero.

Total Two question I have

1. How to prove it is cyclic?

2. How to prove it is cyclic without galios theory?

Answer 1

You're right that $K$ has $4$ elements.

If you're talking about the additive group of $K$ being cyclic, then that is not correct. Since the field $F_2$ is of characteristic $2$, every element of $K$ satisfies $x+x=0$, so the additive group of $K$ is just the Klein $4$ group.

If you're talking about the multiplicative group $K^*$, then yes, $K=\{0,a,b,c\}$ for some nonzero elements $a,b,c$, and $0$ is just the zero of the additive group. Then $\{a,b,c\}$ is a group of order $3$, but there is only one group of order $3$, namely the cyclic group of order $3$.