Tutorial 3 : Separable Extensions and Finite Fields


1. Let $ f(x)=x^{p^n}-a \in F[x]$ where $ n$ is a positive integer and $ F$ is a field of characteristic $ p.$ Show that $ f(x)$ is irreducible $ F[x]$ if and only if $ a \notin F^p.$



2. Show that $ \bigcap_{i=0}^{\infty} F^{p^i}$ is the largest perfect subfield of $ F.$



3. Identify the finite fields $ \mathbb{Z}[i]/(1+i)$ and $ \mathbb{Z}[i]/(2+i).$



4. Find a necessary and sufficient condition on $ n$ and $ m$ so that $ \mathbb{F}_{p^n}$ is a subfield of $ \mathbb{F}_{p^m}.$


5. Draw the poset of subfields of $ \mathbb{F}_{3^{18}}$



6. Show that the order of the Frobenius automorphism $ \phi : K:=\mathbb{F}_{p^n} \rightarrow \mathbb{F}_{p^n}$ is $ n.$


7. Let $ f(x)$ be irreducible over $ \mathbb{F}_p$ and let it have a root $ r \in \mathbb{F}_{p^n}.$ Show that the roots of $ f(x)$ are precisely $ r, r^p, r^{p^2}, \ldots, r^{p^{m-1}}.$


8. Let $ K$ and $ L$ be subfields of $ \mathbb{F}_{p^n}$ having $ p^s$ and $ p^t$ elements respectively. How many elements does the field $ K \cap L$ have ?