1. T

    For p = 3, find the number of monic irreducible polynomials in...?

    1.) For p = 3, find the number of monic irreducible polynomials in Zp[x] of the following degrees (a) 5 (b) 9 (c) 10 2.) Repeat Exercise 1 for a general prime p. Your answers will be formulas in p. Can anybody help me out, at least with part 1!? There must be any easier way than writing them...
  2. T

    Find the number of monic irreducible polynomials in...?

    Find the number of monic irreducible polynomials in Z2[x] of the following degrees: (a) 5 (b) 9 (c) 10 There has got to be an easier way then writing them out, but I can't figure it out, any help??
  3. S

    Transforming polynomial with rational coeffs to monic with integer coeffs

    I've read there is a way to transform polynomials in Q[x] to monic polynomials in Z[x]. How? Consider the following one i Q[x]: f(x) = Sum{k} a_k/b_k x^k = (Prod b_k)^(-1) Sum{k} [Prod{l =/= k} a_k b_l] x^k. If we are to solve f(x) = 0 we can get rid of the factor so that we are left with...
  4. T

    Find a monic polynomial, deduce that the polynomial is irreducible...

    Set d = sqrt (3) + sqrt (2) in the real numbers R. (a) Find a monic polynomial f(x) in Q[x] of degree 4 that has d as a root. (b) If h, k, l, and m are rational numbers such that hd3+kd2+ld+m=0, substitute sqrt(3) + sqrt(2) for d, expand the result, and collect terms. Deduce that h,k,l, and m...
  5. A

    Finding monic irreducible polynomials.

    13. Find all monic irreducible polynomials of degree \leq 3 over \bbold{Z}_3. The problem before this contains a hint that says to derive a way to tell "at a glance" whether or not a polynomial has a root. I'm just not seeing it, any suggestions would be appreciated. This problem comes from...
  6. P

    Find all monic irreducible polynomicals

    My question is the following: Find all monic irreducible polynomials of degree <= 3 over Z3. Then, using the list write (x^2 - 2x + 1) as a product of irreducible polynomials. Any help would be greatly appreciated. Thanks in advance.
  7. B

    Monic Polynomials

    I want to know how many monic polynomials of degree n in \mathbb{F}_p[x] are that to do not take on the value 0 for x \in \mathbb{F}_p[x] . Can anyone give a hint?
  8. I

    Irreducible Monic Polynomials

    The problem is: Count the number of irreducible monic polynomials of degree 2 in (Z/13Z)[X]. Prove your answer. What would be the best way to do this problem, or any tips?
  9. S

    number of monic irreducible polyynomials

    I need help with irreducible polynomials. Could anybody help me please? The problem is: Find the pattern for the number of monic irreducible polynomials of degree 6 in Z_2[x]! Is it different to look for monic irreducible and irreducible polynomials? Thank you very much
  10. H

    monic polynomials

    Any ideas on this one? Let f(x) be a monic polynomial over Z. Show that if n is a nonzero integer such that f(n) = 0, then n divides f(0). Thanks!
  11. C

    Monic Polynomials

    Show that there is only one way (disregarding the order of the factors) to factor x^2 +x+3 as a product of monic irreducible polynomials in Z(sub5)[x]. So, I found that f(1) produces a zero for this polynomial, then I used to the division algorithm to get (x^2 +x+3)/(x-1) = (x+2). However...
  12. J

    Monic polynomial help

    Find the monic polynomial of degree 5 which has 1, 1 + i and 2 − i as three of its roots.. any help on this question id appreciate.
  13. K

    monic polynomials

    Can somebody give me hints on these question please? (i) List all the monic polynomials over F2 of degree =< 3. (ii) Determine which of these polynomials are irreducible over F2. (iii) Factorize the reducible polynomials into irreducible polynomials. Thanks alot
  14. C

    Monic polynomial

    Looking for help: Suppose f(x) belongs to C[x] is a monic polynomial of degree n with roots c_1, c_2, ..., c_n. Prove that the sum of roots is -a_n-1 and their product is (-1)^na_0.