Orders of Galois Field

Question

Here's what I'm working on:

Find the orders of all the nonzero elements of $GF(2, x^4+x+1)$.

I believe I should approach it the same way I did for the orders of the roots of unity in this, but am not sure what to do. Would a cyclic table come in handy for this type of problem?

Answer 1

The multiplicative group of a finite field is cyclic, in this case of order $15$.

Since, clearly, $x$ has neither order $3$ nor $5$, it must be a generator of the multiplicative group.

So all you need to do is find the canonical form of $x^3, x^6, x^9, x^{12}$ (which have order 5) and $x^5, x^{10}$ (which have order 3), and then everything else (except, of course, $1$) has order 15.