Exercise 2.17 in Rudolf Lidl's Finite Fields big red book.

Question

Ex. 2.17 on page 79 asks the following:

prove that: $$f(t^q)=(f(t))^q \ \ \ \forall f(t)\in \mathbb{F}_q[t]$$ my attempt at solution: Ok then I write a polynomial: $f(t)=a_0+a_1t+\ldots+a_mt^m$ and $f(t^q)=a_0+a_1t^q+\ldots + a_m t^{qm}$ and $(f(t))^q=(a_0+a_1t+\ldots +a_mt^m)^q$.

Now, how to finish this proof?

Thanks!

Answer 1

Hint: Over a field of characteristic $p$ where $p$ is prime, consider how $(x+y)^p$ simplifies using the binomial theorem.