The trace mapping Tr is defined on GF(p), p prime, by
![1679_301cf1b2-883c-4a23-9b13-7cdf4819cf2e.png](https://secure.tutorsglobe.com/CMSImages/1679_301cf1b2-883c-4a23-9b13-7cdf4819cf2e.png)
a) Prove that Tr (x) GF (p), for every
![1679_301cf1b2-883c-4a23-9b13-7cdf4819cf2e.png](https://secure.tutorsglobe.com/CMSImages/1679_301cf1b2-883c-4a23-9b13-7cdf4819cf2e.png)
b) Prove that Tr is a linear mapping (hint: Cor. B.28).
c) Prove that Tr takes on every value in GF(p) equally often.
d) Replace p by q in this problem, where q is a prime power, and verify the same statements.