I am an undergraduate working through Chris Hall's result about infinitely many twin irreducible polynomials over finite fields.
He begins his argument with a lemma,
If $q \equiv 1$ mod $l$ for some odd prime $l,$ where $q$ is an odd prime power, and $\beta \in \mathbb{F}^\times_q - {\mathbb{F}_q^\times}^l,$ then $x^{l^m} - \beta$ is irreducible in $\mathbb{F}_q[x]$ for all $m \geq 0$.
He cites his proof as theorem 9.1 from chapter 6 of Lang's Algebra text, which goes as follows:
Let $k$ be a field and $n$ an integer $\geq 2.$ Let $a \in k, a \neq 0.$ Assume that for all prime numbers p such that $p \mid n$ we have that $a \notin k^p,$ and if $4 \mid n$ then $a \notin -4k^4.$ Then we have that $X^n - a$ is irreducible in $k[X].$
So the first issue is translating Hall's statement and notation to that of Lang's. I fully understand how Hall's condition that $\beta \in \mathbb{F^\times_q - F_q}^{\times l}$ satisfies that of Lang's condition. Since Lang requires that $a \neq 0$ and that $a$ is not a $p$-th power in the field. I think I have interpreted this much correctly. The one part I don't seem to understand is the equivalence relation between $q$ and 1. I don't see how this implies the condition imposed by Lang.
Secondly, assuming that I've interpreted what Hall has written correctly, this claim seems a little weaker than what Lang goes to prove. Since Lang assumes that $n$ is of the form $m \cdot p^r$ where $m$ is prime to $p$ and $p$ is odd, but Hall only assumes that the degree of the polynomial is a prime power (hence m = 1), or did I misinterpret his claim?
Also, I have no real exposure to Galois groups and little exposure to field extensions. If someone could explain, intuitively, what is the Norm between field and field extensions and how it relates to Galois groups?