exercises-poly.ipynb

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Section17.4Exercises

¶

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 ${\mathbb Z}_{12}$

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 ${\mathbb Z}_{12}$

Hint

(a) $9x^2 + 2x + 5\text{;}$ (b) $8x^4 + 7x^3 + 2x^2 + 7x\text{.}$

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 ${\mathbb Z}_2[x]$

Hint

(a) $5 x^3 + 6 x^2 - 3 x + 4 = (5 x^2 + 2x + 1)(x -2) + 6\text{;}$ (c) $4x^5 - x^3 + x^2 + 4 = (4x^2 + 4)(x^3 + 3) + 4x^2 + 2\text{.}$

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 ${\mathbb Z}_{12}$

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 ${\mathbb Z}_2$

Hint

(a) No zeros in ${\mathbb Z}_{12}\text{;}$ (c) 3, 4.

6

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

7

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

Hint

Look at $(2x + 1)\text{.}$

8

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

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

Hint

(a) Reducible; (c) irreducible.

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 ${\mathbb Z}_{10}[x]\text{.}$

Hint

One factorization is $x^2 + x + 8 = (x + 2)(x + 9)\text{.}$

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?

Hint

The integers $\mathbb Z$ do not form a field.

14

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

Hint

False.

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 $R[x] \cong S[x]\text{.}$

Hint

Let $\phi : R \rightarrow S$ be an isomorphism. Define $\overline{\phi} : R[x] \rightarrow S[x]$ by $\overline{\phi}(a_0 + a_1 x + \cdots + a_n x^n) = \phi(a_0) + \phi(a_1) x + \cdots + \phi(a_n) x^n\text{.}$

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 $x - a\text{.}$

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 $a_n \neq 0\text{.}$ 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 $( {\mathbb Z}[x], +)\text{.}$

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

Hint

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 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

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

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 of $f(x)\text{.}$

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 $D(f(x)) = f'(x)\text{.}$

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

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

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.

Hint

Find a nontrivial proper ideal in $F[x]\text{.}$

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 $\deg( p(x) + q(x) ) \leq \max( \deg p(x), \deg q(x) )\text{.}$