Irreducible Polynomials

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kathrynag

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1. Let F be a field with p^n elements. Show that F has a subfield K with p^m elements if
and only if m | n.
2. Let K be a finite field. Show that the product of all the nonzero elements of K is ?1.
3. Write f(x) = x^16 ? x in Z2[x] (that is, f(x) = x^2^4? x) as the product of all monic
irreducible polynomials over Z2 of degree dividing 4.
4. (a) Determine the number of monic irreducible polynomials of degree 6 in Z2[x].
(b) Determine the number of monic irreducible polynomials of degree 11 in Z3[x].



Ideas:
1.I think this may have something to do with degrees
2.I started by considering |k|=p^n. Then I tried looking at x^p^n-x=x(x^(p^n-1)-1).
3.
4. Consider degrees?
 
1) The => direction is simple: Let A be the prime subfield. Then [F:K][K:A] = [F:A] = n. For the other direction, if m | n then x^m-1 divides x^n-1. Though theres more to do after this.
2) Finite products in a commutative group may be reordered.
3) Hmm... Trial and error? What divides 4? 1,2,4.
4) http://en.wikipedia.org/wiki/Finite_fie ... nite_field
 
Not sure what you mean with your hint for 2.
For 3 is there an easier way than just trial and error?
 
If the field is Z_7, we have 0,1,2,3,4,5,6 The product of the nonzero elements is 1*2*3*4*5*6, or we may reorder them as 1*6*(2*4)(3*5) = 1*(-1)*(1)*(1) = -1. Get it now?

I don't know of any easier method. You know that x(x-1) is a factor though.
 
Also x^4+x^3+x^2+x+1,x^4+x^3+1,x^4+x^2+1, x^4+x+1. It's writing as a product that's getting me now.
 
I found a hint for 4 that told me to use this theorem and apply degrees:
Let F=GF(q) where q=p^n. The monic irreducible factors of x^q^n-x in F[x] are precisely the monic irreducible polynomials in F[x] whose degree is a divisor of m.
Note GF=Galois field.
I just am a little confused on using this theorem.
 
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