Factorization of Polynomials

Consider the polynomial $f(x)=3x^{2}-6$. Which of the following integral domains is $f(x)$ irreducible over?

1. $\mathbb{Z}$

2. $\mathbb{Q}$

3. $\mathbb{R}$

Let $f(x)\in\mathbb{Z}[x]$. Prove or disprove the following statements.

1. If $f(x)$ is reducible over $\mathbb{Z}$ then it is reducible over $\mathbb{Q}$.

2. If $f(x)$ is reducible over $\mathbb{Q}$ then it is reducible over $\mathbb{Z}$.

Prove that, for all prime numbers $p$, there exists a non-negative integer $n<p$ such that $n^{2}+2n+1\equiv 0\pmod{p}$.

Let $n$ be an arbitrary positive integer. Find a polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ that is irreducible over $\mathbb{Q}$.

Let $f(x)=x^{4}+3$.

1. Show that $f(x)$ is irreducible over $\mathbb{Q}$.

2. Explain why $x^{4}+4x^{3}+6x^{2}+4x+4$ has no zeroes in $\mathbb{Z}$.

3. Show that $x^{4}-8x^{3}+24x^{2}-32x+19$ is irreducible over $\mathbb{Q}$.

Show that the map $\phi$ given in 1.26 is indeed a ring automorphism.

Prove 1.33.

Let $p(x)=x^{2}+2$, $f(x)=2x+3$, and $g(x)=4x^{2}+3x+2$ be polynomials in $\mathbb{Z}_{5}[x]$.

1. Show that $p(x)$ is irreducible over $\mathbb{Z}_{5}$.

2. Hence construct a field, $F$, of 25 elements.

3. Let $I=\left\langle p(x)\right\rangle$ be an ideal of $F$. Simplify the following cosets.

   1. $(f(x)+I)+(g(x)+I)$

   2. $(f(x)+I)(g(x)+I)$

Fill in the remaining of the multiplication table in 1.35.

Determine which of the following polynomials, if any, are irreducible over $\mathbb{Q}$.

1. $f_{1}(x)=x^{3}+x^{2}+x+1$

2. $f_{2}(x)=6x^{3}+x^{2}+x+1$

3. $f_{3}(x)=x^{3}+9$

4. $f_{4}(x)=x^{4}+5$

5. $f_{5}(x)=x^{4}-2x^{3}+x^{2}-x+1$

6. $f_{6}(x)=x^{4}+12x^{3}+54x^{2}+108x+86$

By considering the polynomial $x^{2}-2$, prove that $\sqrt{2}$ is irrational. You may assume $\sqrt{2}\in\mathbb{R}$.

Prove that any primitive degree 1 polynomial in $\mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$.

Let $f(x)=x^{3}+6$.

1. Show that $f(x)$ is reducible over $\mathbb{Z}_{7}$.

2. Write $f(x)$ as a product of irreducible polynomials over $\mathbb{Z}_{7}$.

Find all distinct irreducible polynomials of

1. degree 2

2. degree 3

in $\mathbb{Z}_{2}[x]$.

Let $p$ be a prime number. Show that the number of reducible polynomials over $\mathbb{Z}_{p}$ with the form $x^{2}+ax+b$, where $a,b\in\mathbb{Z}_{p}$, is $\frac{p(p+1)}{2}$.

We show a failure case of the Mod $p$ Irreducibility Test (1.18). Let the polynomial $f(x)=x^{4}+1$.

1. Prove that $f(x)$ is irreducible over $\mathbb{Q}$.

2. Show that $f(x)$ is reducible over $\mathbb{Z}_{2}$ via factorization.For the rest of this part, assume $p>2$.

   1. Suppose there is an $r\in\mathbb{Z}_{p}$ satisfying $r^{2}=k$ for some $k$ (to be specified). Deduce a factorization of $f(x)$ in the following cases.

      1. $k=2$

      2. $k=-1$

      3. $k=-2$

   2. Let $\mathbb{Z}_{p}^{\ast}=\mathbb{Z}_{p}\setminus\{0\}$. Explain why $\mathbb{Z}_{p}^{\ast}$ is a group under multiplication…