Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$).
What I currently have:
$0,1$ are clearly roots of $f$.
Suppose $a,b$ are roots of $f$. Then $(a+b)^{...Six Degrees: The Science of a Connected Age (2004 in paperback, ISBN 0-393-32542-3 and 2003 in hardcover, ISBN 0-393-04142-5) is a popular science book by Duncan J. Watts covering the application of network theory to sociology.
The book covers Watts' own work on small-world networks, and continues on to cover scale-free networks, network searching, epidemics and network failures, social decision making, thresholds in networks, and innovation in large organizations and its lack.
In addition to covering the theoretical models and empirical case studies, the book also includes several stories about...
I have an assignment question,
Let $F$ be a field of characteristics $p$ prime
(b) Show that roots of $X^{p^r}-X$ is a subfield of $F$ with, with $p^s$ elements where $s \le r$. (Hint example: assume F is finite field with $4$ elements, take $p=2, r=3$ and compute $s$.
The part to show th...
