# Thread: show that its an irreducible polynomial

1. ## show that its an irreducible polynomial

Hello

Consider the two finite fields $\displaystyle \mathbb{F}_m$ and $\displaystyle \mathbb{F}_n$ and let n be a prime number, q be a generator of $\displaystyle \mathbb{F}_n^*$.

Now show that the polynomial $\displaystyle \sum_{k=0}^{n-1} X^k$ is irreducible over $\displaystyle \mathbb{F}_m$ -->

Well if I define $\displaystyle f(X):=\sum_{k=0}^{n-1} X^k$ then I think I'll have to assume that f(X)=h(X)*g(X) over $\displaystyle \mathbb{F}_m$ with degree(h)>0 and degree(g)>0 ? But how does one do that?

q has order n-1. What does that mean? Doesn't that mean that f(q) has order n-1 as well? How can I prove this?

2. ## Re: show that its an irreducible polynomial

You're missing some information on $\displaystyle m$. For example if K is the splitting field for f then $\displaystyle K = \mathbb{F}_{n^t}$ as it is a field extension of $\displaystyle \mathbb{F}_n$. So I think the theorem will hold if you maintain that $\displaystyle (m,n)=1$.