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Abstract Algebra problem (related to Rings of Polynomails

  1. Dec 9, 2008 #1
    We'll I made it through another semester, but it seems that I am completely stuck on the last problem of the last homework assignment. I've made a little progress, but I'm really having trouble understanding the question. Perhaps someone on these forums will have some insight

    1. The problem statement, all variables and given/known data
    Let F be a field. An element [tex] \phi \in F^F [/tex] is a polynomial function on F if there exists an [tex] f(x) \in F[x] \text{ s.t. } \phi(a) = f(a) \; \forall a \in F [/tex]

    a) Show that the set [tex] P_F [/tex] of all polynomial functions on F form a subring of [tex]F^F[/tex]

    b) Show that the ring [tex] P_F [/tex] is not necessarily isomorphic to F[x] (Hint: show that if F is a finite field, then [tex] P_F [/tex] and F[x] don't have the same number of elements)

    2. Relevant equations
    In this problem, [tex] F^F [/tex] refers to the set of all functions mapping F to F, and F[x] refers to the ring of polynomials with coefficients in F.


    3. The attempt at a solution

    Part b) doesn't seem so difficult. If we let [tex]F = \mathbb{Z}_2 [/tex], then there are only 4 functions in all since there are only two possible values in the domain and range. But [tex] \mathbb{Z}_2 [x] [/tex] has infinitely many elements, so they cannot be isomorphic.

    Part a) though has me stumped. We have that [tex]P_F \subseteq F^F[/tex], so for any [tex]\phi \in P_F, \longrightarrow \phi: F \rightarrow F [/tex]. If we let [tex] \phi, \psi \in P_F \; f(x), g(x) \in F[x][/tex], then we have that [tex] \phi(a) = f(a), \; \psi(a) = \g(a) [/tex].

    Am I understanding this correctly? How can I demonstrate that it satisfies the ring axioms? Just directly apply them and see if it works? Or am I missing something?
     
  2. jcsd
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