/sci/ - Science & Math

<span class="math">f(x)=x^n+a_{1}x^{n-1}+....+a_(n-1)x ^n +a_n[/spoiler] a polynomial with integer coefficients. Also let p be a prime. Prove that there are at most n numbers x in the set S={0,1,2,3,.....,p-1} for which f(x) is divisible by p. I really dont know where to start

>>6380378

Maybe consider the sum of the mods of the individual terms?

x^n = A mod p, where A is nonzero, since x is coprime to p.

pshhh this is highschool tier stuff, just move the x's to one side and add a 1 to the other. remember to transform the final values of x into its .99999.. form(quantum mechanics tells us that a photon behaves like a particle in some situations and as a wave in others. so, if you hypothesize of a cat concealed in a box, the cat can either be either state UNTIL an observer opens the box and checks it. This is called the wave-particle duality, and can be applied to numbers as well - meaning each number has two possible states.)

>>6380405
but if we consider all the sums and all of them will be of the form =Amodp, with A non zero, doesnt it mean there is the remainder left? dont we want it to be zero(the remainder)?

>>6380476
What'd the remainder of 2 mod 2?

Or if A = kp then its remainder will be zero still

>>6380480

Yeah, you want to show that the upper bound for this happening is n.

You might have to break it down and consider first
x^n mod p, x^(n-1) mod p, ...
then consider
a1*x(n-1) mod p?

Also, since x is coprime to p, I think the collection of x^n, x^(n-1), ..., x form a group, which could be helpful in the proof.

>>6380486

Also, since p is prime, S is a field, right? Maybe that could help?

>>6380495

This might help:

http://www.mth.msu.edu/~jhall/classes/codenotes/PolyAlg.pdf

>>6380434