What is the Theorem regarding the number of polynomial zeros modulo p and H?

[Kontilera]

Hello I am currently learning some of the basics of number theory, and struggling to understand this Theorem. Could someone please explain it with maby a simple example? :)

THRM:(Number of polynomial zero mod p and H)
Let p be a prime number and let H be a polynomial that is irruducible modulo p. Furthermore let P be a polynomial that has degree d>=0 modulo p. Then P has at most d polynomial zeros that are pairwise not congruent modulo p and H.

 

[Mark44]

Then P has at most d polynomial zeros that are pairwise not congruent modulo p and H.

What does the part in bold mean?
I understand what "congruent mod p" means, but I don't understand what "congruent mod p and H" means.

 

[Kontilera]

okey, i will explain with an example. say P=X^2 and Q=2X^3+X-2, and p=3, dividing P by H we obtain a rest of +4.
Then dividing Q by H we obtain a rest of -5.
+4 is congurent to +1 mod 3, and -5 is congurent with +1 mod 3.
We now say that the polynomials P is congurent to Q mod(p,H). Because they have the same remainders when dividing by H mod n.

 

[Mark44]

okey, i will explain with an example. say P=X^2 and Q=2X^3+X-2, and p=3, dividing P by H we obtain a rest of +4.

You haven't said what H is.

Then dividing Q by H we obtain a rest of -5.
+4 is congurent to +1 mod 3, and -5 is congurent with +1 mod 3.
We now say that the polynomials P is congurent to Q mod(p,H). Because they have the same remainders when dividing by H mod n.

 

[Kontilera]

srry!
H is an non-constant polynomial whoose leading coeff is coprime to n.

 

[fresh_42]

May I summarize:
We have a prime ##p## and polynomials ##P(x), H(x) ∈ ℤ[x]## where ##\deg P = d ## and ##H[x] \mod p ## is irreducible in ##Z_p[x]##.
Then ##d \mod p ≥ 0##. But this is always the case.
Now we have to show that ##P(x)## has at most ##d## zeros ##\{x_1,...,x_d\}## in ##ℤ## or in ##ℤ_p##?
Or did you mean ## \{ x-x_1,...,x-x_d \} ## as "polynomial zeros"?

Those are pairwise incongruent "modulo ##(p,H)##" which you defined as follows:
Two polynomials ##P(x),Q(x) ∈ ℤ[x]## are congruent modulo ##(p,H)## if ##\frac{P}{H} = \frac{Q}{H} \mod n##.
I suppose ##n=p##? Or ##n=d##? And the division of the polynomials is performed in which Ring? Or shall we divide ##\frac{P(x_i)}{H(x_i)}##?