Written Assignment No. 4

due Monday, November 21, 2005

General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.

Assigned Exercises

1. Let A be a domain. Prove that the group of units in the polynomial ring A[x] is isomorphic to the group of units in A.

2. Let F be a finite field with |F | = q. Let R be the set of all functions F ⟶ F. Observe that the number |R| of such functions is q^{q}. Observe that R is an abelian group under pointwise addition of functions, i.e.,

(f + g)(x) = f(x) + g(x) ,

and that R becomes a ring when multiplication is defined pointwise, i.e.,

(f · g)(x) = f(x) g(x) .

(a) What ring homomorphism ϕ : F [x] → R has the properties that (i) ϕ applied to a polynomial of degree 0 is the corresponding constant function and (ii) ϕ applied to the polynomial x is the identity function?

(b) Find a polynomial of degree q in the kernel of ϕ.

(c) What is the kernel of ϕ?

(d) Show that ϕ is surjective.