General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.
Let be a domain. Prove that the group of units in the polynomial ring is isomorphic to the group of units in .
Let be a finite field with . Let be the set of all functions . Observe that the number of such functions is . Observe that is an abelian group under pointwise addition of functions, i.e., and that becomes a ring when multiplication is defined pointwise, i.e.,
What ring homomorphism has the properties that (i) applied to a polynomial of degree is the corresponding constant function and (ii) applied to the polynomial is the identity function?
Find a polynomial of degree in the kernel of .
What is the kernel of ?
Show that is surjective.