## Exercises17.5Exercises

###### 1.

List all of the polynomials of degree $$3$$ or less in $${\mathbb Z}_2[x]\text{.}$$

###### 2.

Compute each of the following.

1. $$(5x^2 + 3x - 4) + (4x^2 - x + 9)$$ in $${\mathbb Z}_{12}[x]$$

2. $$(5x^2 + 3x - 4) (4x^2 - x + 9)$$ in $${\mathbb Z}_{12}[x]$$

3. $$(7x^3 + 3x^2 - x) + (6x^2 - 8x + 4)$$ in $${\mathbb Z}_9[x]$$

4. $$(3x^2 + 2x - 4) + (4x^2 + 2)$$ in $${\mathbb Z}_5[x]$$

5. $$(3x^2 + 2x - 4) (4x^2 + 2)$$ in $${\mathbb Z}_5[x]$$

6. $$(5x^2 + 3x - 2)^2$$ in $${\mathbb Z}_{12}[x]$$

###### 3.

Use the division algorithm to find $$q(x)$$ and $$r(x)$$ such that $$a(x) = q(x) b(x) + r(x)$$ with $$\deg r(x) \lt \deg b(x)$$ for each of the following pairs of polynomials.

1. $$a(x) = 5 x^3 + 6x^2 - 3 x + 4$$ and $$b(x) = x - 2$$ in $${\mathbb Z}_7[x]$$

2. $$a(x) = 6 x^4 - 2 x^3 + x^2 - 3 x + 1$$ and $$b(x) = x^2 + x - 2$$ in $${\mathbb Z}_7[x]$$

3. $$a(x) = 4 x^5 - x^3 + x^2 + 4$$ and $$b(x) = x^3 - 2$$ in $${\mathbb Z}_5[x]$$

4. $$a(x) = x^5 + x^3 -x^2 - x$$ and $$b(x) = x^3 + x$$ in $${\mathbb Z}_2[x]$$

###### 4.

Find the greatest common divisor of each of the following pairs $$p(x)$$ and $$q(x)$$ of polynomials. If $$d(x) = \gcd( p(x), q(x) )\text{,}$$ find two polynomials $$a(x)$$ and $$b(x)$$ such that $$a(x) p(x) + b(x) q(x) = d(x)\text{.}$$

1. $$p(x) = x^3 - 6x^2 + 14x - 15$$ and $$q(x) = x^3 - 8x^2 + 21x - 18\text{,}$$ where $$p(x), q(x) \in {\mathbb Q}[x]$$

2. $$p(x) = x^3 + x^2 - x + 1$$ and $$q(x) = x^3 + x - 1\text{,}$$ where $$p(x), q(x) \in {\mathbb Z}_2[x]$$

3. $$p(x) = x^3 + x^2 - 4x + 4$$ and $$q(x) = x^3 + 3 x -2\text{,}$$ where $$p(x), q(x) \in {\mathbb Z}_5[x]$$

4. $$p(x) = x^3 - 2 x + 4$$ and $$q(x) = 4 x^3 + x + 3\text{,}$$ where $$p(x), q(x) \in {\mathbb Q}[x]$$

###### 5.

Find all of the zeros for each of the following polynomials.

1. $$5x^3 + 4x^2 - x + 9$$ in $${\mathbb Z}_{12}[x]$$

2. $$3x^3 - 4x^2 - x + 4$$ in $${\mathbb Z}_{5}[x]$$

3. $$5x^4 + 2x^2 - 3$$ in $${\mathbb Z}_{7}[x]$$

4. $$x^3 + x + 1$$ in $${\mathbb Z}_2[x]$$

###### 6.

Find all of the units in $${\mathbb Z}[x]\text{.}$$

###### 7.

Find a unit $$p(x)$$ in $${\mathbb Z}_4[x]$$ such that $$\deg p(x) \gt 1\text{.}$$

###### 8.

Which of the following polynomials are irreducible over $${\mathbb Q}[x]\text{?}$$

1. $$\displaystyle x^4 - 2x^3 + 2x^2 + x + 4$$

2. $$\displaystyle x^4 - 5x^3 + 3x - 2$$

3. $$\displaystyle 3x^5 - 4x^3 - 6x^2 + 6$$

4. $$\displaystyle 5x^5 - 6x^4 - 3x^2 + 9 x - 15$$

###### 9.

Find all of the irreducible polynomials of degrees $$2$$ and $$3$$ in $${\mathbb Z}_2[x]\text{.}$$

###### 10.

Give two different factorizations of $$x^2 + x + 8$$ in $${\mathbb Z}_{10}[x]\text{.}$$

###### 11.

Prove or disprove: There exists a polynomial $$p(x)$$ in $${\mathbb Z}_6[x]$$ of degree $$n$$ with more than $$n$$ distinct zeros.

###### 12.

If $$F$$ is a field, show that $$F[x_1, \ldots, x_n]$$ is an integral domain.

###### 13.

Show that the division algorithm does not hold for $${\mathbb Z}[x]\text{.}$$ Why does it fail?

###### 14.

Prove or disprove: $$x^p + a$$ is irreducible for any $$a \in {\mathbb Z}_p\text{,}$$ where $$p$$ is prime.

###### 15.

Let $$f(x)$$ be irreducible in $$F[x]\text{,}$$ where $$F$$ is a field. If $$f(x) \mid p(x)q(x)\text{,}$$ prove that either $$f(x) \mid p(x)$$ or $$f(x) \mid q(x)\text{.}$$

###### 16.

Suppose that $$R$$ and $$S$$ are isomorphic rings. Prove that $$R[x] \cong S[x]\text{.}$$

###### 17.

Let $$F$$ be a field and $$a \in F\text{.}$$ If $$p(x) \in F[x]\text{,}$$ show that $$p(a)$$ is the remainder obtained when $$p(x)$$ is divided by $$x - a\text{.}$$

###### 18.The Rational Root Theorem.

Let

\begin{equation*} p(x) = a_n x^n + a_{n - 1}x^{n - 1} + \cdots + a_0 \in \mathbb Z[x]\text{,} \end{equation*}

where $$a_n \neq 0\text{.}$$ Prove that if $$p(r/s) = 0\text{,}$$ where $$\gcd(r, s) = 1\text{,}$$ then $$r \mid a_0$$ and $$s \mid a_n\text{.}$$

###### 19.

Let $${\mathbb Q}^*$$ be the multiplicative group of positive rational numbers. Prove that $${\mathbb Q}^*$$ is isomorphic to $$( {\mathbb Z}[x], +)\text{.}$$

###### 20.Cyclotomic Polynomials.

The polynomial

\begin{equation*} \Phi_n(x) = \frac{x^n - 1}{x - 1} = x^{n - 1} + x^{n - 2} + \cdots + x + 1 \end{equation*}

is called the cyclotomic polynomial. Show that $$\Phi_p(x)$$ is irreducible over $${\mathbb Q}$$ for any prime $$p\text{.}$$

###### 21.

If $$F$$ is a field, show that there are infinitely many irreducible polynomials in $$F[x]\text{.}$$

###### 22.

Let $$R$$ be a commutative ring with identity. Prove that multiplication is commutative in $$R[x]\text{.}$$

###### 23.

Let $$R$$ be a commutative ring with identity. Prove that multiplication is distributive in $$R[x]\text{.}$$

###### 24.

Show that $$x^p - x$$ has $$p$$ distinct zeros in $${\mathbb Z}_p\text{,}$$ for any prime $$p\text{.}$$ Conclude that

\begin{equation*} x^p - x = x(x - 1)(x - 2) \cdots (x - (p - 1))\text{.} \end{equation*}
###### 25.

Let $$F$$ be a field and $$f(x) = a_0 + a_1 x + \cdots + a_n x^n$$ be in $$F[x]\text{.}$$ Define $$f'(x) = a_1 + 2 a_2 x + \cdots + n a_n x^{n - 1}$$ to be the derivative of $$f(x)\text{.}$$

1. Prove that

\begin{equation*} (f + g)'(x) = f'(x) + g'(x)\text{.} \end{equation*}

Conclude that we can define a homomorphism of abelian groups $$D : F[x] \rightarrow F[x]$$ by $$D(f(x)) = f'(x)\text{.}$$

2. Calculate the kernel of $$D$$ if $$\chr F = 0\text{.}$$

3. Calculate the kernel of $$D$$ if $$\chr F = p\text{.}$$

4. Prove that

\begin{equation*} (fg)'(x) = f'(x)g(x) + f(x) g'(x)\text{.} \end{equation*}
5. Suppose that we can factor a polynomial $$f(x) \in F[x]$$ into linear factors, say

\begin{equation*} f(x) = a(x - a_1) (x - a_2) \cdots ( x - a_n)\text{.} \end{equation*}

Prove that $$f(x)$$ has no repeated factors if and only if $$f(x)$$ and $$f'(x)$$ are relatively prime.

###### 26.

Let $$F$$ be a field. Show that $$F[x]$$ is never a field.

###### 27.

Let $$R$$ be an integral domain. Prove that $$R[x_1, \ldots, x_n]$$ is an integral domain.

###### 28.

Let $$R$$ be a commutative ring with identity. Show that $$R[x]$$ has a subring $$R'$$ isomorphic to $$R\text{.}$$

###### 29.

Let $$p(x)$$ and $$q(x)$$ be polynomials in $$R[x]\text{,}$$ where $$R$$ is a commutative ring with identity. Prove that $$\deg( p(x) + q(x) ) \leq \max( \deg p(x), \deg q(x) )\text{.}$$