Abstract Algebra: Theory and Applications

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 $\mathrm{deg}r(x)<\mathrm{deg}b(x)$ for each of the following pairs of polynomials.

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

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

3. $a(x)=4x^5-x^3+x^2+4$ 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 $q(x)=x^3+3x-2\text{,}$ where $p(x),q(x)\in {\mathbb{Z}}_5[x]$

4. $p(x)=x^3-2x+4$ and $q(x)=4x^3+x+3\text{,}$ 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 $\mathrm{deg}p(x)>1\text{.}$

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

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 $F[x_1,\dots ,x_n]$ 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

where $a_n\ne 0\text{.}$ 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 ${\mathbb{Q}}^{\ast }$ be the multiplicative group of positive rational numbers. Prove that ${\mathbb{Q}}^{\ast }$ is isomorphic to $(\mathbb{Z}[x],+)\text{.}$

20. Cyclotomic Polynomials.

The polynomial

is called the cyclotomic polynomial. Show that ${\mathrm{\Phi }}_p(x)$ 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

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 $f^{\prime }(x)=a_1+2a_{2}x+\cdots +n{a}_n{x}^{n-1}$ to be the derivative of $f(x)\text{.}$

1. Prove that

   $$(f+g{)}^{\prime }(x)=f^{\prime }(x)+g^{\prime }(x)\text{.}$$

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

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

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

4. Prove that

   $$(fg{)}^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)\text{.}$$

5. Suppose that we can factor a polynomial $f(x)\in F[x]$ into linear factors, say

   $$f(x)=a(x-a_1)(x-a_2)\cdots (x-a_n)\text{.}$$

   Prove that $f(x)$ has no repeated factors if and only if $f(x)$ and $f^{\prime }(x)$ 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 $R[x_1,\dots ,x_n]$ is an integral domain.

28.

Let $R$ be a commutative ring with identity. Show that $R[x]$ has a subring $R^{\prime }$ 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 $\mathrm{deg}(p(x)+q(x))\le max(\mathrm{deg}p(x),\mathrm{deg}q(x))\text{.}$