For an algebra HW, I had to show that any element in a finite field can be written as the sum of two squares of elements in that same field (the problem comes from Judson). I was able to do that. I was wondering if this idea could be (ever, maybe only in certain cases) generalized to the sum of two elements to prime powers.
My first thoughts for beginning to prove this involved using the prime characteristic of a finite field and thinking of something similarly motivated to what is referred to as the "Freshman's Dream" in our textbook: for a finite field with prime characteristic $p$, then for field elements $a, b$ we have $$(a + b)^{p^n} = a^{p^n} + b^{p^n}$$ for all natural numbers $n$.
Obviously I can't use the theorem for any prime I want to. I was hoping there might be something else of note to prove/disprove this seamlessly. Does this only work for the prime that is our field's characteristic, or is there more to this?