Finite field

[PURE MATH]

1. Show that for $a \in \mathbb{F}_q $ and $ n \in \mathbb{N} $ the polynomial $x^{q^n}-x+na$ is divisible by $x^q-x+a$ over $ \mathbb{F}_q. $
2. Let $ \mathbb{F}_q$ be a finite field of characteristics $p$. Prove that $f \in \mathbb{F}_q[x] $ satisfies $f'(x)=0$ if and only if $f$ is the $p$th power of some polynomial in $\mathbb{F}_q[x] .$

ช่วยหน่อยครับ เริ่มไม่เป็นเลย

[MINGA]

1. Using $x^q = x$, we get $x^q-x+a = a$ and $x^{q^n}-x+na = na$ in $\mathbb{F}_q.$


2. Fact: $(ax+by)^p = ax^p+by^p$ for $a,b\in\mathbb{F}_q$ where $q$ is a power of $p.$

Consider $\frac{d}{dx}(a_k x^k) = ka_k x^k = 0$ iff $p|k$ or $p|a_k.$

Let $f(x)=\sum_{k=1}^n a_k x^k \in\mathbb{F}_q[x].$ Since $f'(x)=0$, $a_k=0$ for all $p\nmid k.$

So, $f(x) = \sum_{i=1}^m a_{pi}x^{pi} = (\sum_{i=1}^m a_{ki}x^{i})^p.$