Chapter 22

In Exercises 1 through 4 , find the sum and the product of the given polynomials in the given polynomial ring. 1\. \(f(x)=4 x-5, g(x)=2 x^{2}-4 x+2\) in \(Z_{8}[x]\).

\(f(x)=2 x^{2}+3 x+4, g(x)=3 x^{2}+2 x+3\) in \(Z_{6}[x]\).

How many polynomials are there of degree \(\leq 3\) in \(Z_{2}[x] ?\) (Include 0 .)

\(x^{5}+3 x^{3}+x^{2}+2 x\) in \(Z_{5}\)

\(f(x) g(x)\) where \(f(x)=x^{3}+2 x^{2}+5\) and \(g(x)=3 x^{2}+2 x\) in \(\mathrm{Z}_{7}\) \\}

Consider the element $$ f(x, y)=\left(3 x^{3}+2 x\right) y^{3}+\left(x^{2}-6 x+1\right) y^{2}+\left(x^{4}-2 x\right) y+\left(x^{4}-3 x^{2}+2\right) $$ of \((Q[x])[y]\). Write \(f(x, y)\) as it would appear if viewed as an element of \((Q[y])[x]\).

. Find a polynomial of degree \(>0\) in \(Z_{4}[x]\) that is a unit.

Mark each of the following true or false. a. The polynomial \(\left(a_{n} x^{n}+\cdots+a_{1} x+a_{0}\right) \in R[x]\) is 0 if and only if \(a_{i}=0\), for \(i=0,1, \cdots, n\). b. If \(R\) is a commutative ring, then \(R[x]\) is commatative. c. If \(D\) is an integral domain, then \(D[x]\) is an integral domain. d. If \(R\) is a ring containing divisors of 0 , then \(R[x]\) has divisors of \(0 .\) e. If \(R\) is a ring and \(f(x)\) and \(g(x)\) in \(R[x]\) are of degrees 3 and 4 , respectively, then \(f(x) g(x)\) may be of degree 8 in \(R[x]\). f. If \(R\) is any ring and \(f(x)\) and \(g(x)\) in \(R[x]\) are of degrees 3 and 4 , respectively, then \(f(x) g(x)\) is always of degree \(7 .\) g. If \(F\) is a subfield \(E\) and \(\alpha \in E\) is a zero of \(f(x) \in F[x]\), then \(\alpha\) is a zero of \(h(x)=f(x) g(x)\) for all \(g(x) \in F[x]\). \(\mathbf{h}\). If \(F\) is a field, then the units in \(F[x]\) are precisely the units in \(F\). i. If \(R\) is a ring, then \(x\) is never a divisor of 0 in \(R[x]\). j. If \(R\) is a ring, then the zero divisors in \(R[x]\) are precisely the zero divisors in \(R\). Theory

Prove that if \(D\) is an integral domain, then \(D[x]\) is an integral domain.

Prove the left distributive law for \(R[x]\), where \(R\) is a ring and \(x\) is an indeterminate.

Let \(F\) be a field of characteristic zero and let \(D\) be the formal polynomial differentiation map, so that $$ D\left(a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2} x^{n}\right)=a_{1}+2 \cdot a_{2} x+\cdots+n \cdot a_{n} x^{n-1} $$ a. Show that \(D: F \mid x] \rightarrow F[x]\) is a group homomorphism of \(\langle F[x],+)\) into itself. Is \(D\) a ring homomorphism? b. Find the kernel of \(D\). c. Find the image of \(F[x]\) under \(D\).