Factorization of Polynomials

Verify the assertion made in Example 2 (the polynomial $f(x) = 2x^{2}+4$ is irreducible over $\mathbb {R}$ but reducible over $\mathbb {C}$).

Suppose that $D$ is an integral domain and $F$ is a field containing $D$. If $f(x) \in D[x]$ and $f(x)$ is irreducible over $F$ but reducible over $D$, what can you say about the factorization of $f(x)$ over $D$?

Show that a nonconstant polynomial from $\mathbb {Z}^{x}$ that is irreducible over $\mathbb {Z}$ is primitive. (This exercise is referred to in this chapter.)

Suppose that $f(x) = x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0} \in \mathbb {Z}^{x}$. If $r$ is rational and $x-r$ divides $f(x)$, show that $r$ is an integer.

Let $F$ be a field and let $a$ be a nonzero element of $F$.

Let $F$ be a field and $f(x) \in F[x]$. Show that, as far as deciding upon the irreducibility of $f(x)$ over $F$ is concerned, we may assume that $f(x)$ is monic. (This assumption is useful when one uses a computer to check for irreducibility.)

Suppose there is a real number $r$ with the property that $r+1/r$ is an odd integer. Prove that $r$ is irrational.

Show that the equation $x^{2}-7y^{2} = 3$ has no solutions in the integers.

Explain how the Mod $p$ Irreducibility Test (Theorem 17.3) can be used to test members of $\mathbb {Q}^{x}$ for irreducibility.

Suppose that $f(x) \in Z_{p}[x]$ and $f(x)$ is irreducible over $Z_{p}$, where $p$ is a prime. If $\deg f(x) = n$, prove that $Z_{p}[x]/ \left\langle f(x)\right\rangle$ is a field with $p^{n}$ elements.

Construct a field of order $25$.

Construct a field of order $27$.

Show that $x^{3}+x^{2}+x+1$ is reducible over $\mathbb {Q}$. Does this fact contradict the corollary to Theorem 17.4?

Corollary (Irreducibility of $p$th Cyclotomic Polynomial): For any prime $p$, the $p$th cyclotomic polynomial $\Phi_{p}(x) = (x^{p}-1)/(x-1) = x^{p-1}+x^{p-2}+\cdots +x+1$ is irreducible over $\mathbb {Q}$.

Determine which of the polynomials below is (are) irreducible over $\mathbb {Q}$.

Referring to the field in Example 10 and denoting a coset by its representative only, find the representatives for $(x^{2}+x)^{2}$ and $(x^{2}+x)^{-1}$.

Let $F$ denote the field in Example 10 and $F^{*}$ be the multiplicative subgroup of $F$. Prove that for any element $a$ of $F^{*}$ other than the identity, the cyclic subgroup generated by $a$ is $F^{*}$. Would the same be true for a field $F$ if $|F| = 32$? What about $|F| = 16$?

What can you conclude about a polynomial $f(x)$ in $\mathbb {Z}^{x}$ if $f(x)$ is irreducible over $Z_{p}[x]$ for some prime $p$ (i.e., when the coefficients of $f(x)$ are reduced modulo $p$)? What can you conclude about a polynomial $f(x)$ in $\mathbb {Z}^{x}$ if $f(x)$ is reducible over $Z_{p}[x]$ for every prime $p$ (i.e., when the coefficients of $f(x)$ are reduced modulo $p$)?

Let $f(x) \in Z_{2}[x]$ and $\deg f(x) = 5$. If neither $0$ nor $1$ is a zero of $f(x)$, explain why, in order to prove that $f(x)$ is irreducible over $Z_{2}$, it suffices to show that $x^{2}+x+1$ is not a factor of $f(x)$.

For the field $Z_{7}[x]/I$ where $I = \left\langle x^{2}+2\right\rangle$, find the multiplicative orders of $x+I$ and $x+1+I$. Find the multiplicative inverse of $x+I$.

Let $F$ be a field and $f(x) \in F[x]$ be reducible over $F$ with $\deg f(x) > 1$. Prove that $F[x]/ \left\langle f(x)\right\rangle$ is not an integral domain.

Show that $x^{4}+1$ is irreducible over $\mathbb {Q}$.

Let $f(x) = ax^{4}+bx^{2}+c \in \mathbb {Z}^{x}$. If $p(x)$ is a factor of $f(x)$ in $\mathbb {Z}^{x}$ prove that $p(-x)$ is also a factor of $f(x)$ in $\mathbb {Z}^{x}$.

Let $f(x) = x^{3}+6 \in Z_{7}[x]$. Write $f(x)$ as a product of irreducible polynomials over $Z_{7}$.

Let $f(x) = x^{3}+x^{2}+x+1 \in Z_{2}[x]$. Write $f(x)$ as a product of irreducible polynomials over $Z_{2}$.

Prove that $x^{4}+15x^{3}+7$ is irreducible over $\mathbb {Q}$.

Find all zeros of $f(x) = 3x^{2}+x+4$ over $Z_{7}$ by substitution. Find all zeros of $f(x)$ by using the quadratic formula $(-b \pm \sqrt{b^{2}-4ac}) \cdot (2a)^{-1}$ (all calculations are done in $Z_{7}$). Do your answers agree? Should they? Find all zeros of $g(x) = 2x^{2}+x+3$ over $Z_{5}$ by substitution. Try the quadratic formula on $g(x)$. Do your answers agree? State necessary and sufficient conditions for the quadratic formula to yield the zeros of a quadratic from $Z_{p}[x]$, where $p$ is a prime greater than $2$.

Let $p$ be a prime.

Let $p$ be a prime.

Show that for every prime $p$ there exists a field of order $p^{2}$.

Prove that, for every positive integer $n$, there are infinitely many monic polynomials of degree $n$ in $\mathbb {Z}^{x}$ that are irreducible over $\mathbb {Q}$.

Show that the field given in Example 11 in this chapter is isomorphic to the field given in Example 9 in Chapter 13.

Let $f(x) \in Z_{p}[x]$. Prove that if $f(x)$ has no factor of the form $x^{2}+ax+b$, then it has no quadratic factor over $Z_{p}$.

Find all monic irreducible polynomials of degree $2$ over $Z_{3}$.

Given that $\pi$ is not the zero of a nonzero polynomial with rational coefficients, prove that $\pi^{2}$ cannot be written in the form $a\pi +b$, where $a$ and $b$ are rational.

(Rational Root Theorem) Let

and $a_{n} \neq 0$. Prove that if $r$ and $s$ are relatively prime integers and $f(r/s) = 0$, then $r \mid a_{0}$ and $s \mid a_{n}$.

Let $F$ be a field and let $p(x), a_{1}(x), a_{2}(x), \ldots , a_{k}(x) \in F[x]$, where $p(x)$ is irreducible over $F$. If $p(x) \mid a_{1}(x)a_{2}(x) \cdots a_{k}(x)$, show that $p(x)$ divides some $a_{i}(x)$. (This exercise is referred to in the proof of Theorem 17.6.)

Theorem 16.4 ($F[x]$ Is a PID): Let $F$ be a field. Then $F[x]$ is a principal ideal domain.

Let $F$ be a field and $p(x) \in F[x]$. Use Theorem 14.4 to prove that if $\left\langle p(x)\right\rangle$ is a maximal ideal in $F[x]$, then $p(x)$ is irreducible over $F$ (see Theorem 17.5).

If $p$ is a prime, prove that $x^{p-1}-x^{p-2}+x^{p-3}-\cdots -x+1$ is irreducible over $\mathbb {Q}$.

Let $F$ be a field and let $p(x)$ be irreducible over $F$. If $E$ is a field that contains $F$ and there is an element $a$ in $E$ such that $p(a) = 0$, show that the mapping $\varphi : F[x] \rightarrow E$ given by $f(x) \rightarrow f(a)$ is a ring homomorphism with kernel $\left\langle p(x)\right\rangle$. (This exercise is referred to in Chapter 19.)

Prove that the ideal $\left\langle x^{2}+1\right\rangle$ is prime in $\mathbb {Z}^{x}$ but not maximal in $\mathbb {Z}^{x}$.

Let $F$ be a field and let $p(x)$ be irreducible over $F$. Show that $\left\{ a+ \left\langle p(x)\right\rangle \mid a \in F\right\}$ is a subfield of $F[x]/ \left\langle p(x)\right\rangle$ isomorphic to $F$. (This exercise is referred to in Chapter 19.)

Let $F$ be a field and let $f(x)$ be a polynomial in $F[x]$ that is reducible over $F$. Prove that $\left\langle f(x)\right\rangle$ is not a prime ideal in $F[x]$.

Example 1 in this chapter shows the converse of Theorem 17.2 is not true. That is, a polynomial $f(x)$ in $\mathbb {Z}^{x}$ can be reducible over $\mathbb {Z}$ but irreducible over $\mathbb {Q}$. State a condition on $f(x)$ that makes the converse true.