General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.
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.
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) . |
What ring homomorphism phi : F [x] --> R has the properties that (i) phi applied to a polynomial of degree 0 is the corresponding constant function and (ii) phi applied to the polynomial x is the identity function?
Find a polynomial of degree q in the kernel of phi.
What is the kernel of phi?
Show that phi is surjective.