Composition series of $\mathbb F_{p^k}$ I am trying to find the composition series of the group $\mathbb F_{p^k}$, where $p$ is prime , and $k\ge 1$. 
From Jordan Hölder equation it has length $k$ , i am quite confused , 
It looks quite natural to write $${e}\subset \mathbb F_p \subset...........\mathbb F_{p^k}$$
And each of them are definitely the maximal subgroups ( my doubt is how do i follow normality of every subgroup and the abelian nature of factor groups ie $\mathbb F_{p^i}/\mathbb F_{p^{i-1}}$ . 
Thank you . 
 A: $\mathbb{F}_{p^k}$ is a finite field, which means it has an additive group and a multiplicative group.  You can think of elements of $\mathbb{F}_{p^k}$ as polynomials $a_0+a_1x+\ldots +a_{k-1}x^{k-1}$, where the $a_i\in \mathbb{Z}_p$ and $x$ is an indeterminate element.  (It's easy to see combinatorially that there are $p^k$ polynomials of this form.)  The addition is done just like polynomial addition, that is, $$\left(a_0+a_1x+\ldots +a_{k-1}x^{k-1}\right)+\left(b_0+b_1x+\ldots +b_{k-1}\alpha^{k-1}\right)\\=(a_0+b_0)+(a_1+b_1)x+\ldots +(a_{k-1}+b_{k-1})x^{k-1}.$$
The multiplication is more complicated.  When you make $\mathbb{F}_{p^k}$, you do it by picking some polynomial $m(x)$ of degree $k$ which is irreducible over the polynomial ring $\mathbb{Z}_p[x]$.  Then you 'mod out' by $m(x)$ in  $\mathbb{Z}_p[x]$.  What this means is that when you multiply two elements $p(x)$ nd $q(x)$ of $\mathbb{F}_{p^k}$ written in the form given above, you take the remainder of $p(x)q(x)$ modulo $m(x)$.  This is done simply by polynomial long division, which is exactly the same as normal long devision - you just keep dividing by $m(x)$ until you can't anymore, and whatever is left over is $p(x)q(x)$ in the desired polynomial form.
Anyway, that's a little introduction to finite fields to help you on your way.  You can see that both the addition and multiplication are commutative operations.  If you look up the definition of normality, I am sure this will ease your concerns.  If you'd like some advice though, I think you should go back and relearn the basic definitions of all this stuff before you try to read about advanced things like Jordan-Holder (which, by the way, doesn't just apply to groups).  It's going to be difficult to solve problems if you don't know what the objects you're working with are.
