1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Number theory basics :)

  1. Feb 23, 2016 #1
    Hello im 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.
     
  2. jcsd
  3. Feb 24, 2016 #2

    Mark44

    Staff: Mentor

    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.
     
  4. Feb 24, 2016 #3
    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.
     
  5. Feb 24, 2016 #4

    Mark44

    Staff: Mentor

    You haven't said what H is.
     
  6. Feb 24, 2016 #5
    srry!
    H is an non-constant polynomial whoose leading coeff is coprime to n.
     
  7. Feb 24, 2016 #6

    fresh_42

    Staff: Mentor

    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)}##?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Number theory basics :)
  1. Number Theory (Replies: 7)

  2. Number theory (Replies: 2)

  3. Number Theory (Replies: 1)

  4. Basics of numbers! (Replies: 29)

  5. Number Theory (Replies: 2)

Loading...