Polynomial Rings

Let $f(x) = 4x^{3}+2x^{2}+x+3$ and $g(x) = 3x^{4}+3x^{3}+3x^{2}+x+4$, where $f(x), g(x) \in Z_{5}[x]$. Compute $f(x)+g(x)$ and $f(x) \cdot g(x)$.

In $Z_{3}[x]$, show that the distinct polynomials $x^{4}+x$ and $x^{2}+x$ determine the same function from $Z_{3}$ to $Z_{3}$.

Show that $x^{2}+3x+2$ has four zeros in $Z_{6}$.

If $R$ is a commutative ring, show that the characteristic of $R[x]$ is the same as the characteristic of $R$.

Prove that Theorem 16.2 and its corollaries are true for $D[x]$ where $D$ is an integral domain and $g(x)$ has a lead coefficient that is a unit.

Theorem 16.2 (Division Algorithm for $F[x]$): Let $F$ be a field and let $f(x), g(x) \in F[x]$ with $g(x) \neq 0$. Then there exist unique polynomials $q(x)$ and $r(x)$ in $F[x]$ such that $f(x) = g(x)q(x)+r(x)$ and either $r(x) = 0$ or $\deg r(x) < \deg g(x)$. Its corollaries: Corollary 1 (Remainder Theorem) -- for $a \in F$ and $f(x) \in F[x]$, $f(a)$ is the remainder in the division of $f(x)$ by $x-a$; Corollary 2 (Factor Theorem) -- $a$ is a zero of $f(x)$ if and only if $x-a$ is a factor of $f(x)$.

List all the polynomials of degree 2 in $Z_{2}[x]$. Which of these are equal as functions from $Z_{2}$ to $Z_{2}$?

Prove that every element in the ring of polynomial functions from $Z_{3}$ to $Z_{3}$ under the usual definition of addition and multiplication can be written in the form $ax^{2}+bx+c$ where $a, b, c \in Z_{3}$. Generalize to $Z_{p}$ where $p$ is prime.

Prove that the ring of polynomial functions with coefficients from $Z_{3}$ and the usual definition of addition and multiplication is not an integral domain. Why does this statement not contradict Theorem 16.1?

Theorem 16.1 ($D$ an Integral Domain Implies $D[x]$ an Integral Domain): If $D$ is an integral domain, then $D[x]$ is an integral domain.

Write $x^{3}+x^{2}-2$ as a product of linear factors in $Z_{3}[x]$.

For any positive integer $n$, how many polynomials are there of degree $n$ over $Z_{2}$? How many distinct polynomial functions from $Z_{2}$ to $Z_{2}$ are there?

Let $f(x) = 5x^{4}+3x^{3}+1$ and $g(x) = 3x^{2}+2x+1$ in $Z_{7}[x]$. Determine the quotient and remainder upon dividing $f(x)$ by $g(x)$.

Find a polynomial with real coefficients that has $i$ and $2+i$ as zeros.

If $\varphi : R \rightarrow S$ is a ring homomorphism, define $\bar{\varphi }: R[x] \rightarrow S[x]$ by $(a_{n}x^{n}+\cdots +a_{0}) \rightarrow \varphi (a_{n})x^{n}+\cdots +\varphi (a_{0})$. Show that $\bar{\varphi }$ is a ring homomorphism. (This exercise is referred to in Chapters 17 and 31.)

If the rings $R$ and $S$ are isomorphic, show that $R[x]$ and $S[x]$ are isomorphic. (The converse is not true.)

Prove that $Z_{4}[x]$ has infinitely many units and infinitely many nilpotent elements.

Let $f(x)$ and $g(x)$ be cubic polynomials with integer coefficients such that $f(a) = g(a)$ for four integer values of $a$. Prove that $f(x) = g(x)$. Generalize.

Show that the polynomial $2x+1$ in $Z_{4}[x]$ has a multiplicative inverse in $Z_{4}[x]$.

Show that Theorem 16.4 is false for any commutative ring that has a zero divisor.

Theorem 16.4 (used here as a comparison point): $F[x]$ is a principal ideal domain whenever $F$ is a field.

(Degree Rule) Let $D$ be an integral domain and $f(x), g(x) \in D[x]$. Prove that $\deg (f(x) \cdot g(x)) = \deg f(x)+\deg g(x)$. Show, by example, that for a commutative ring $R$ it is possible that $\deg f(x)g(x) < \deg f(x)+\deg g(x)$, where $f(x)$ and $g(x)$ are nonzero elements in $R[x]$. (This exercise is referred to in this chapter, Chapter 17, and Chapter 18.)

Prove that the only idempotent elements in $Z_{4}[x]$ are $0$ and $1$.

Let $f(x)$ belong to $F[x]$, where $F$ is a field. Let $a$ be a zero of $f(x)$ of multiplicity $n$, and write $f(x) = (x-a)^{n}q(x)$. If $b \neq a$ is a zero of $q(x)$, show that $b$ has the same multiplicity as a zero of $q(x)$ as it does for $f(x)$. (This exercise is referred to in this chapter.)

Prove that $x^{3}-x^{2}-4$ has a zero in every field.

Let $R$ be a commutative ring. For any $r \in R$ prove that the mapping from $R[x]$ to $R$ given by $\varphi (f(x)) = f(r)$ is a ring homomorphism. This mapping is called the evaluation homomorphism.

If $f(x) \in \mathbb {C}^{x}$ and $f(i) = 0$ prove that $f(i^{3}) = 0$.

For which primes $p$ is $x-2$ a factor of $f(x) = x^{4}-2x-2$ over $Z_{p}$?

Let $b$ be a unit in a commutative ring. If $b$ is a zero of $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$, prove that $b^{-1}$ is a zero of $a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}$.

Prove that for any prime $p$ the group $U(p)$ is cyclic.

Theorem 16.3 (Polynomials of Degree $n$ Have at Most $n$ Zeros): a polynomial of degree $n$ over a field has at most $n$ zeros, counting multiplicity.

Show that the ring $Z_{4}[x]$ has infinitely many elements with multiplicative order $2$.

Give an example of a commutative ring that has a polynomial of degree 1 that has five distinct zeros.

Let $n > 1$ be a non-prime integer. Show that there is a polynomial $f(x) \in Z_{n}[x]$ of degree $n-1$ that has $n$ distinct zeros. (Compare with Theorem 16.3, which says a polynomial of degree $n$ over a field has at most $n$ zeros, counting multiplicity -- the point of this exercise being that $Z_n$ is not a field when $n$ is not prime.)

For any field $F$, recall that $F(x)$ denotes the field of quotients of the ring $F[x]$. Prove that there is no element in $F(x)$ whose square is $x$.

Give an example of a polynomial in $Z_{5}[x]$ of positive degree that has the property that $f(a) = 1$ for all $a \in Z_{5}$.

Let $D$ be an integral domain. Prove that no nonconstant element in $D[x]$ can have a multiplicative inverse in $D[x]$.

Write $x^{3}+2x^{2}-x-2$ as a product of linear polynomials over $Z_{3}$.

Prove that $x^{3}-2x^{2}-9$ has a zero in every field.

Let $F$ be a field and $f(x)$ and $g(x)$ be nonzero elements in $F[x]$. Prove that $\left\langle f(x)\right\rangle = \left\langle g(x)\right\rangle$ if and only if $f(x) = cg(x)$ for some $c$ in $F$.

Let $F$ be a field and let $I$ be the ideal $\left\{ f(x) \in F[x] \mid f(1) = 0 \text{ and } f(2) = 0\right\}$. Find an element $g(x) \in F[x]$ such that $I = \left\langle g(x)\right\rangle$.

If the only nilpotent element in $Z_{n}$ is $0$, prove that no element in $Z_{n}[x]$ of positive degree is an idempotent.

Determine all zeros of $x^{9}+1$ in $Z_{17}[x]$.

Prove that $\mathbb {Q}^{x}/ \left\langle x^{2}+1\right\rangle$ isomorphic to the subfield of complex numbers $\mathbb {Q}(i) = \left\{ a+bi \mid a, b \in \mathbb {Q}\right\}$.

Show that $x+1$ is a factor of $f(x) = x^{25}+1$ in $Z_{37}$ but $(x+1)^{2}$ is not.

Suppose the $f(x) \in Z_{5}[x]$ is a non-constant polynomial that has the property that $f(a) = 2$ for all $a \in Z_{5}$. What is the minimum possible degree for $f(x)$?

Let $F$ be a field, and let $f(x)$ and $g(x)$ belong to $F[x]$ and not both zero. If there is no polynomial of positive degree in $F[x]$ that divides both $f(x)$ and $g(x)$, prove that there exist polynomials $h(x)$ and $k(x)$ in $F[x]$ with the property that $f(x)h(x)+g(x)k(x) = 1$. (This exercise is referred to in Chapter 19.)

If $F$ is a field and $f(x)$ and $g(x)$ belong to $F[x]$ and are not both zero, a polynomial $d(x)$ in $F[x]$ is said to be a greatest common divisor of $f(x)$ and $g(x)$ if $d(x)$ divides both $f(x)$ and $g(x)$ and $d(x)$ has maximum degree among all such polynomials. Prove that for any $f(x)$ and $g(x)$ belonging to $F[x]$, not both zero, $\left\langle f(x), g(x)\right\rangle = \left\langle d(x)\right\rangle$, where $d(x)$ is a greatest common divisor of $f(x)$ and $g(x)$. Prove that for any $f(x)$ and $g(x)$ belonging to $F[x]$, not both zero, there is a unique monic greatest common divisor of $f(x)$ and $g(x)$.

In $\mathbb {Z}^{x}$, prove that the ideal $I = \left\langle x, 2\right\rangle$ is not a principal ideal.

Let $F$ be an infinite field and let $f(x), g(x) \in F[x]$. If $f(a) = g(a)$ for infinitely many elements $a$ of $F$, show that $f(x) = g(x)$.

Let $F$ be a field and let $p(x) \in F[x]$. If $f(x), g(x) \in F[x]$ and $\deg f(x) < \deg p(x)$ and $\deg g(x) < \deg p(x)$, show that $f(x)+ \left\langle p(x)\right\rangle = g(x)+ \left\langle p(x)\right\rangle$ implies $f(x) = g(x)$. (This exercise is referred to in Chapter 19.)

Find a polynomial with integer coefficients that has $1/2$ and $-1/3$ as zeros.

Prove that for any fixed positive integer $n$, the number of elements in a field that have multiplicative order at most $n$ is at most $1+2+\cdots +n$.

Let $f(x) \in \mathbb {R}^{x}$. Suppose that $f(a) = 0$ but $f'(a) \neq 0$, where $f'(x)$ is the derivative of $f(x)$. Show that $a$ is a zero of $f(x)$ of multiplicity $1$.

Show that Corollary 2 of Theorem 16.2 is true over any commutative ring with unity.

Corollary 2 (Factor Theorem): Let $F$ be a field, $a \in F$, and $f(x) \in F[x]$. Then $a$ is a zero of $f(x)$ if and only if $x-a$ is a factor of $f(x)$.

Show that Theorem 16.3 is true for polynomials over integral domains.

Theorem 16.3 (Polynomials of Degree $n$ Have at Most $n$ Zeros): a polynomial of degree $n$ over a field has at most $n$ zeros, counting multiplicity.

Let $F$ be a field and let

Show that $I$ is an ideal of $F[x]$ and find a generator for $I$.

Let $F$ be a field and let $f(x) = a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0} \in F[x]$. Prove that $x-1$ is a factor of $f(x)$ if and only if $a_{n}+a_{n-1}+\cdots +a_{0} = 0$.

For any prime $p$ prove that the ring $Z_{p^{2}}[x]$ has infinitely many units.

Find infinitely many polynomials $f(x)$ in $Z_{3}[x]$ such that $f(a) = 0$ for all $a$ in $Z_{3}$.

For every prime $p$, show that

in $Z_{p}[x]$.

(Wilson's Theorem) For every integer $n > 1$, prove that $(n-1)! \bmod n = n-1$ if and only if $n$ is prime.

For every prime $p$, show that $(p-2)! \bmod p = 1$.

Find the remainder upon dividing $98!$ by $101$.

Find an element $a$ in $Z_{5}$ such that $x+4$ is a factor of $x^{48}+x^{21}+a$.

Let $\varphi$ be the ring homomorphism from $\mathbb {Q}^{x}$ to $\mathbb {Q}$ given by $\varphi (f(x)) = f(1)$. Find the monic polynomial $g(x)$ in $\mathbb {Q}^{x}$ such that $\operatorname {Ker} \varphi = \left\langle g(x)\right\rangle$. To which familiar ring is $\mathbb {Q}^{x}/\operatorname {Ker} \varphi$ isomorphic?

Give an example of a field that properly contains the field of complex numbers $\mathbb {C}$.

If $I$ is an ideal of a ring $R$, prove that $I[x]$ is an ideal of $R[x]$.

Give an example of a commutative ring $R$ with unity and a maximal ideal $I$ of $R$ such that $I[x]$ is not a maximal ideal of $R[x]$.

Let $R$ be a commutative ring with unity. If $I$ is a prime ideal of $R$, prove that $I[x]$ is a prime ideal of $R[x]$.

Show that $1$ is the only solution of $x^{25}-1 = 0$ in $Z_{37}$.

Prove that $\mathbb {Q}^{x}/ \left\langle x^{2}-2\right\rangle$ is ring-isomorphic to $\mathbb {Q}(\sqrt{2}) = \left\{ a+b\sqrt{2} \mid a, b \in \mathbb {Q}\right\}$.

Let $f(x) \in \mathbb {R}^{x}$. If $f(a) = 0$ and $f'(a) = 0$, show that $(x-a)^{2}$ divides $f(x)$.

Let $F$ be a field and let $I = \left\{ f(x) \in F[x] \mid f(a) = 0 \text{ for all } a \text{ in } F\right\}$. Prove that $I$ is an ideal in $F[x]$. Prove that $I$ is infinite when $F$ is finite and $I = \left\{ 0\right\}$ when $F$ is infinite. When $F$ is finite, find a monic polynomial $g(x)$ such that $I = \left\langle g(x)\right\rangle$.

Let $g(x)$ and $h(x)$ belong to $\mathbb {Z}^{x}$ and let $h(x)$ be monic. If $h(x)$ divides $g(x)$ in $\mathbb {Q}^{x}$, show that $h(x)$ divides $g(x)$ in $\mathbb {Z}^{x}$. (This exercise is referred to in Chapter 31.)

Let $R$ be a ring and $x$ be an indeterminate. Prove that the rings $R[x]$ and $R[x^{2}]$ are ring-isomorphic.

Let $f(x)$ be a nonconstant element of $\mathbb {Z}^{x}$. Prove that $f(x)$ takes on infinitely many values in $\mathbb {Z}$.

Let $F$ be a field. Find a generator for the ideal $I = \left\{ f(x) \in F(x) \mid f(0) = 0 \text{ and } f(1) = 0\right\}$. Generalize.

Suppose that $F$ is a field and there is a ring homomorphism from $\mathbb {Z}$ onto $F$. Show that $F$ is isomorphic to $Z_{p}$ for some prime $p$.

Let $p$ be a prime, $b$ belong to $Z_{p}$, and $f(x)$ belong to $Z_{p}[x]$. Prove that if $f(b) = 0$, then $f(b^{p}) = 0$. Generalize.

Suppose $f(x)$ is a polynomial with odd integer coefficients and even degree. Prove that $f(x)$ has no rational zeros.

Find the remainder when $x^{51}$ is divided by $x+4$ in $Z_{7}[x]$.

Let $F$ be a field. Show that there exist $a, b \in F$ with the property that $x^{2}+x+1$ divides $x^{43}+ax+b$.