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Section Polynomial Rings
Worksheet Part 1: Polynomial Rings & Division
The goal of these problems is to explore practice applying the division algorithm, improve your familiarity with properties of polynomial rings, and practice constructing homomorphisms of polynomial rings.
1. Use the Division Algorithm to find the quotient and remainder in
\(\Z_5[x]\) upon dividing
\(4x^4-x^3+x^2+4\) by
\(x^3-2\text{.}\)
2. Show that
\(2x+1\) has a multiplicative inverse in
\(\Z_4[x]\text{.}\)
3. Are there any non-constant polynomials in
\(\Z[x]\) that have a multiplicative inverse? If so, find them.
4. Prove the
Degree Rule : If
\(D\) is an integral domain, then for any nonzero
\(f,g \in D[x]\) we have
\(\deg(f\cdot g)=\deg (f) +\deg(g).\) Give an example of polynomials
\(f,g\) in some commutative
\(R[x]\) where
\(\deg(f\cdot g)\lt\deg (f)+\deg(g)\)
5.
The map \(\phi: \Z_3\to\Z_6\) given by \(\phi(x)=4x\) is a ring homomorphism. Show that \(\overline{\phi}: \Z_3[x]\to \Z_6[x]\) given by
\begin{equation*}
\overline{\phi}\left(\sum_{i=0}^n a_ix^i\right)=\sum_{i=0}^n \phi(a_i)x^i
\end{equation*}
is a ring homomorphism.
6. What does your argument above suggest as a generalization?
