Construct a finite field of 16 elements and find a generator for its multiplicative group. Find all generators of multiplicative group.
Very obvious Construction of a field with 16 elements according to me would be ${\mathbb{F_2}[x]}/{f(x)}$ where $f(x)$ is an irreducible polynomial of order 4 in $\mathbb{F_2}[x]$. I took $f(x)=x^4+x+1$.Elements of the resulting field are
$\{0,1,x,x^2,x^3,1+x,1+x^2,1+x^3,x+x^2,x+x^3,x^2+x^3,x+x^2+x^3,1+x+x^2,1+x+x^3,1+x+x^3,1+x^2+x^3,1+x+x^2+x^3\}$.
Now, Question is to find a generator of its multiplicative group.
I have calculated by hand that $<x>$ is the whole multiplicative group. That was very obvious by intuition that $x$ generates. Now the question is what other elements generate that group.
I have tried for $x^2$, though i did not calculate whole group,with in few steps i found $x\in<x^2>$ so i concluded $x^2$ generates multiplicative group.
In case of $1+x$ , with in 4/5 steps i got multiplicative identity 1. So $1+x$ does not generate this group.
It would become a mess if i write by hand what subgroup would each element generate?
Is there any better way to find out what all elements generate the multiplicative group?
Best Answer
If $\langle x\rangle$ is the whole multiplicative group, it is isomorphic to $C_{15}$, so your generators will be $x^a$ for all $a$ satisfying $\gcd(a,15)=1$. There should be $\phi(15)=8$ such values.