Consider the polynomials $f(x),g(x) \in \mathbb{Q}[x]$ given by $f(x) = -4x^4 - 3x^2 + 2x + 4$ and $g(x) = 2x^2 + 2x + 1$. Find the quotient and remainder when $f(x)$ is divided by $g(x)$. Show your work.
For a ring $R$, let $U(R)$ denote the group of units of $R$ under multiplication. For a field $F$, we have $U(F) = F^*$ because every nonzero element is a unit. Recall that for a finite field $F$, the group $F^*$ is cyclic.
Find a generator of the group $\mathbb{Z}_7^*$.
Find a generator of the group $U(\mathbb{Z}_8)$, or else show that it is not cyclic.
Find a generator of the group $U(\mathbb{Z}_{10})$, or else show that it is not cyclic.
Consider the polynomial $f(x) = 2x^3 + 2x^2 + 4x + 3$ in $\mathbb{Z}_5[x]$. Express $f(x)$ as a product of irreducible polynomials in $\mathbb{Z}_5[x]$. In your answer, write every coefficient as a number $0$, $1$, $2$, $3$, or $4$; in particular, do not use any minus signs.
Hint: does your polynomial have any zeroes in $\mathbb{Z}_5$? If so, you can factor out a linear term. If not, use Theorem 23.10. We didn't discuss this theorem in the lecture, but it is easy enough that you should know it.
Let $f(x) \in \mathbb{Z}[x]$, say $f(x) = \sum_{i=0}^n a_ix^i$ where $a_n \ne 0$.
Let $p$ and $q$ be nonzero integers with $\gcd(p,q) = 1$. Prove that if the rational number $p/q$ is a zero of $f$, then $p$ divides $a_0$ and $q$ divides $a_n$.
Hint: write out what $f(p/q) = 0$ means, and then multiply by $q^n$.
Assuming that $f(x)$ is monic (meaning that its leading coefficient is $1$,) use part a to prove that every rational zero of $f(x)$ is an integer.
Let $m$ be a positive integer that is not a square. Use part b to prove that $\sqrt{m}$ is irrational.
For each of the following subsets of the polynomial ring $\mathbb{Q}[x]$, say which case applies: (a) it is an ideal, (b) it is a subring but not an ideal, or (c) it is not a subring.
In case (a), you do not need to write your justification. In case (b), say why it is not an ideal. In case (c), say why it is not a subring.
$\{f(x) \in \mathbb{Q}[x] : \deg f(x) \le 0\}$ (in other words, $\mathbb{Q}$ itself.)
$\{f(x) \in \mathbb{Q}[x] : \deg f(x) \le 1\}$
$\{f(x) \in \mathbb{Q}[x] : \deg f(x) \ge 1 \text{ or } f(x) = 0\}$
$\{f(x) \in \mathbb{Q}[x] : f(x) \text{ has only integer coefficients}\}$ (in other words, $\mathbb{Z}[x]$.)
$\{f(x) \in \mathbb{Q}[x] : (x^2 + 1) \mid f(x)\}$, where "$\mid$" means "divides".
$\{f(x) \in \mathbb{Q}[x] : f(x) \text{ is monic}\}$.
Let $R$ and $R'$ be rings and let $\phi : R \to R'$ be a homomorphism.
Let $N'$ be an ideal of $R'$. Prove that the inverse image $\phi^{-1}[N']$ is an ideal of $R$.
Let $N$ be an ideal of $R$. Must the image $\phi[N]$ be an ideal of $R'$? Prove it or give a counterexample.