monic

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.