I've solved the second part by second hypothesis of induction.

alpha + beta= p+1
this implies that alpha+beta is not divisible by p...........(1)

and alpha x beta=1 .............(2)


STEP1 -- To prove that P(1) is not divisible by p.
alpha^1+beta^1=alpha+beta is not divisible by p.[from (1)]

To prove that P(2) is not divisible by p.
alpha^2+beta^2=(alpha+beta)^2 - 2 .alpha beta is not divisible by p

STEP2(INDUCTION ASSUMPTION) -- Let P(k) and P(k-1) be true.
so, alpha^k+beta^k is not divisible by p. ............(3)
and alpha^(k-1)+beta^(k-1) is not divisible by p..............(4)

STEP3 -- To prove that P(k+1) is not divisible by p.
P(k+1)=alpha^(k+1)+beta^(k+1)
=(alpha^k+beta^k)(alpha+beta) - alpha^kbeta -beta.alpha^k
=(alpha^k+beta^k)(alpha+beta) - alpha.beta{alpha^(k-1)+beta^(k-1)}


from(INDUCTION ASSUMPTION)
alpha^k+beta^k is not divisible by p............(3)
and alpha^(k-1)+beta^(k-1) is not divisible by p..............(4)

I'm stuck at this point.
then, how to prove (alpha^k+beta^k)(alpha+beta) - alpha.beta{alpha^(k-1)+beta^(k-1)} is not divisible by p.

I've solved this question many times but haven't proved it perfectly.