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Abstract Algebra: An Introduction

Thomas Hungerford

Chapter 5

Congruence in f[x] and Congruence-Class Arithmetic - all with Video Answers

Educators

Section 1

Congruence in F[x] and Congruence Classes

02:26

Problem 1

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Let $f(x), g(x), p(x) \in F[x]$, with $p(x)$ nonzero. Determine whether $f(x) \equiv g(x)$ $(\bmod p(x))$. Show your work.
(a) $f(x)=x^{5}-2 x^{4}+4 x^{3}+x+1 ; g(x)=3 x^{4}+2 x^{3}-5 x^{2}-9$; $p(x)=x^{2}+1 ; F=\mathbb{Q}$
(b) $f(x)=x^{4}+x^{2}+x+1 ; g(x)=x^{4}+x^{3}+x^{2}+1$; $p(x)=x^{2}+x ; F=\mathbb{Z}_{2}$
(c) $f(x)=3 x^{5}+4 x^{4}+5 x^{3}-6 x^{2}+5 x-7$; $g(x)=2 x^{5}+6 x^{4}+x^{3}+2 x^{2}+2 x-5 ; p(x)=x^{3}-x^{2}+x-1 ; F=\mathbb{R}$

Naman Kumar
Naman Kumar
Numerade Educator
00:45

Problem 2

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
If $p(x)$ is a nonzero constant polynomial in $F[x]$, show that any two polynomials in $F[x]$ are congruent modulo $p(x)$.

Jeyasree R T
Jeyasree R T
Numerade Educator
04:11

Problem 3

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
How many distinct congruence classes are there modulo $x^{3}+x+1$ in $\mathbb{Z}_{2}[x]$ ? List them.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
01:36

Problem 4

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Show that, under congruence modulo $x^{3}+2 x+1$ in $\mathbb{Z}_{3}[x]$, there are exactly 27 distinct congruence classes.

Thomas Emment
Thomas Emment
Numerade Educator
01:33

Problem 5

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Show that there are infinitely many distinct congruence classes modulo $x^{2}-2$ in $Q[x]$. Describe them.

Kenwa Nandi
Kenwa Nandi
Numerade Educator
00:34

Problem 6

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Let $a \in F$. Describe the congruence classes in $F[x]$ modulo the polynomial $x-a$.

Allison Knapp
Allison Knapp
Numerade Educator
04:11

Problem 7

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Describe the congruence classes in $F[x]$ modulo the polynomial $x$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
00:45

Problem 8

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Prove or disprove: If $p(x)$ is relatively prime to $k(x)$ and $f(x) k(x) \equiv g(x) k(x)$ $(\bmod p(x))$, then $f(x) \equiv g(x)(\bmod p(x))$.

Jeyasree R T
Jeyasree R T
Numerade Educator
03:05

Problem 9

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Prove that $f(x) \equiv g(x)(\bmod p(x))$ if and only if $f(x)$ and $g(x)$ leave the same remainder when divided by $p(x)$.

Gregory Higby
Gregory Higby
Numerade Educator
01:29

Problem 10

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Prove or disprove: If $p(x)$ is irreducible in $F[x]$ and $f(x) g(x) \equiv 0_{F}(\bmod p(x))$, then $f(x) \equiv 0_{F}(\bmod p(x))$ or $g(x) \equiv 0_{F}(\bmod p(x))$.

Chris Trentman
Chris Trentman
Numerade Educator
00:45

Problem 11

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
If $p(x)$ is reducible in $F[x]$, prove that there exist $f(x), g(x) \in F[x]$ such that $f(x) \neq 0_{F}(\bmod p(x))$ and $g(x) \not \equiv 0_{F}(\bmod p(x))$ but $f(x) g(x) \equiv 0_{F}(\bmod p(x))$.

Jeyasree R T
Jeyasree R T
Numerade Educator
02:38

Problem 12

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
If $f(x)$ is relatively prime to $p(x)$, prove that there is a polynomial $g(x) \in F[x]$ such that $f(x) g(x) \equiv 1_{F}(\bmod p(x))$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:07

Problem 13

$F$ denotes a field and $p(x)$ a nonzero polynomial in $F[x]$.
Suppose $f(x), g(x) \in \mathbb{R}[x]$ and $f(x) \equiv g(x)(\bmod x)$. What can be said about the graphs of $y=f(x)$ and $y=g(x)$ ?

James Kiss
James Kiss
Numerade Educator