##### 1

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

##### 2

Compute each of the following.

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

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

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

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

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

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

##### 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 $d(x) = \gcd( p(x), q(x) )$, find two polynomials $a(x)$ and $b(x)$ such that $a(x) p(x) + b(x) q(x) = d(x)$.

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

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

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

4. $p(x) = x^3 - 2 x + 4$ and $q(x) = 4 x^3 + x + 3$, 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 ${\mathbb Z}_{12}$

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

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

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

##### 6

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

##### 7

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

##### 8

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

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

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

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

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

##### 9

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

##### 10

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

##### 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 ${\mathbb Z}[x]$. Why does it fail?

##### 14

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

##### 15

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

##### 16

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

##### 17

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

##### 18The 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],\end{equation*} where $a_n \neq 0$. Prove that if $p(r/s) = 0$, where $\gcd(r, s) = 1$, then $r \mid a_0$ and $s \mid a_n$.

##### 19

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

##### 20Cyclotomic 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 $p$.

##### 21

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

##### 22

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

##### 23

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

##### 24

Show that $x^p - x$ has $p$ distinct zeros in ${\mathbb Z}_p$, for any prime $p$. Conclude that \begin{equation*}x^p - x = x(x - 1)(x - 2) \cdots (x - (p - 1)).\end{equation*}

##### 25

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

1. Prove that \begin{equation*}(f + g)'(x) = f'(x) + g'(x).\end{equation*} Conclude that we can define a homomorphism of abelian groups $D : F[x] \rightarrow F[x]$ by $D(f(x)) = f'(x)$.

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

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

4. Prove that \begin{equation*}(fg)'(x) = f'(x)g(x) + f(x) g'(x).\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).\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 $R$.

##### 29

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