Written Assignment No. 4

Assigned Exercises

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

2. Let $F$ be a finite field with $\left|F\right|=q$. Let $R$ be the set of all functions $F\to F$. Observe that the number $\left|R\right|$ of such functions is $q^q$. Observe that $R$ is an abelian group under pointwise addition of functions, i.e., $$\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\text{,}$$ and that $R$ becomes a ring when multiplication is defined pointwise, i.e., $$\left(f·g\right)\left(x\right)=f\left(x\right)g\left(x\right)\text{.}$$

   1. What ring homomorphism $\varphi :F\left[x\right]\to R$ has the properties that (i) $\varphi$ applied to a polynomial of degree $0$ is the corresponding constant function and (ii) $\varphi$ applied to the polynomial $x$ is the identity function?

   2. Find a polynomial of degree $q$ in the kernel of $\varphi$.

   3. What is the kernel of $\varphi$?

   4. Show that $\varphi$ is surjective.