TITLE: Condition implying rationality of $u^n+v^n$
QUESTION [5 upvotes]: $Given :\ u+v \   is \ rational, \ u^2 + v^2 =1 \ , prove \ v^n + u^n \ is \ rational$.
What I have done so far is proving  that $uv$  is rational by expanding $(u+v)^2$.
I expanded $(u+v)^n$ using binomial expansion but I do not think this is the proper way.
Any hint about how to prove it please?

REPLY [3 votes]: Related to another answer and comment, but perhaps less "magic" and more motivated.
It is given that $u+v$ is rational, and you have proved that $uv$ is rational.  By the binomial theorem,
$$\eqalign{u^n+v^n
  &=(u+v)^n-\binom n1u^{n-1}v-\binom n2u^{n-2}v^2-\cdots
    -\binom n{n-2}u^2v^{n-2}-\binom n{n-1}uv^{n-1}\cr
  &=(u+v)^n-\binom n1(uv)(u^{n-2}+v^{n-2})
    -\binom n2(uv)^2(u^{n-4}+v^{n-4})-\cdots\ .\cr}$$
Now every term on the RHS is either rational by what you know already, or can be assumed rational by induction.