Problem 2 Factor the indicated polynomial ... [FREE SOLUTION]

Chapter 8: Problem 2

Factor the indicated polynomial $f(x)$ completely into irreducible factors in the polynomial ring $F[x]$ for the indicated field $F$. $$ f(x)=x^{4}+1 \quad F=\mathbb{Z}_{2} $$

Short Answer

Expert verified

The irreducible factorization of $f(x) = x^4 + 1$ over $\mathbb{Z}_2$ is $(x^2 + x + 1)^2$.

Step by step solution

Understanding the Field and Polynomial

We are given the polynomial $f(x) = x^4 + 1$ and we need to factor it over the finite field $\mathbb{Z}_2$, which consists of two elements: 0 and 1. In $\mathbb{Z}_2$, addition and multiplication are performed modulo 2.

Expressing the Polynomial

Start with the given polynomial $f(x) = x^4 + 1$. Since we know the operations are taken mod 2, we re-write $f(x)$ as $x^4 + 1$ and consider potential factorizations in reduced form, explicitly in terms of simple polynomials.

Attempting Simple Factorization

Try to factor $f(x) = x^4 + 1$ by assuming it can be decomposed into two quadratic polynomials: $(x^2 + ax + b)(x^2 + cx + d)$. Expand this product to match terms with $x^4 + 1$.

Expanding and Matching

Expand the assumed factorization:\[(x^2 + ax + b)(x^2 + cx + d) = x^4 + (a+c)x^3 + (ac+b+d)x^2 + (ad+bc)x + bd.\]Match the coefficients with $x^4 + 0x^3 + 0x^2 + 0x + 1$ and set the equations equal mod 2.

Solving the Equations

To match $x^4 + 1$, set the following:1. $a + c = 0$ (for no $x^3$ term)2. $ac + b + d = 0$3. $ad + bc = 0$4. $bd = 1$Solve these equations modulo 2 to find suitable values for $a, b, c,$ and $d$.

Choosing Possible Values

From equation $bd = 1$, both $b$ and $d$ must be 1 (because any element squared in $\mathbb{Z}_2$ equals 0 or 1, and only 1 squared stays 1). Substitute these values back to solve for $a$ and $c$ from remaining equations.

Final Factorization

Substitute $b = 1$ and $d = 1$ into the remaining system of equations:1. $a + c = 0$2. $ac + 1 + 1 = 0$ simplifies to $ac = 0$3. Both $ad + bc = 0$ is satisfied with these choices.This gives (try values): $a = 0$ and $c = 0$ or vice versa.Thus, the factorization is:\[f(x) = (x^2 + x + 1)(x^2 + x + 1).\]Checking by expansion confirms correctness in $\mathbb{Z}_2$, completing factorization.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

finite fields

Finite fields are mathematical structures with a finite number of elements, where you can perform addition, subtraction, multiplication, and division (except by zero) with certain rules. One commonly used finite field is denoted as $ \mathbb{Z}_p $, where $ p $ is a prime number. This set contains numbers from 0 up to $ p-1 $.

Addition and multiplication in finite fields are performed modulo $ p $.
For example, $ \mathbb{Z}_2 $ is a finite field with two elements: 0 and 1.
In $ \mathbb{Z}_2 $, the operations use modulo 2 arithmetic, meaning 1 + 1 = 0 and 1 鈰� 1 = 1.

This property of having a limited number of elements and operations defined in a specific way makes finite fields a critical concept when dealing with polynomials over fields like $ \mathbb{Z}_2 $. They are frequently used in coding theory and cryptography.

irreducible polynomials

An irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-trivial polynomials in a given field. In terms of analogy, they are like the prime numbers of polynomials.

For a polynomial to be irreducible in a field, there should be no polynomial factors other than itself and 1.
In finite fields, determining if a polynomial is irreducible often involves checking potential factors among lower-degree polynomials.
The factorization of polynomials into irreducible factors is a crucial operation in simplifying expressions and solving polynomial equations.

In our exercise, we factor the polynomial $ f(x) = x^4 + 1 $ into irreducible factors over $ \mathbb{Z}_2 $, which helps in understanding its structure within this particular finite field. This understanding contributes significantly to fields such as algebraic coding theory.

modular arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, known as the modulus. Imagine a clock that goes back to 0 after reaching 12; that's the essence of modular arithmetic.

In equations, it is expressed with the modulo operation, like "$ a \mod m $," resulting in the remainder when $ a $ is divided by $ m $.
Operations performed under modular arithmetic follow specific rules where counting resets at the modulus.
For example, in $ \mathbb{Z}_2 $, entire operations like addition and multiplication adhere to $ x \mod 2 $.

Understanding modular arithmetic is essential when dealing with polynomial rings over finite fields, as it dictates how polynomial coefficients behave. Working within this system, all intermediate arithmetic during calculations respect these modular constraints.

Z2 polynomial ring

A $ \mathbb{Z}_2 $ polynomial ring involves polynomials whose coefficients belong to the field $ \mathbb{Z}_2 $. These polynomials are quite different from those over real numbers due to the limited field coefficients.

In $ \mathbb{Z}_2[x] $, polynomials are formed using coefficients 0 or 1.
All operations, namely addition and multiplication, are conducted using modulo 2 arithmetic.
This ring is not a field but aids in constructing polynomial functions over finite fields like $ \mathbb{Z}_2 $.

The focus of our original exercise was to factor a polynomial $ x^4 + 1 $ in the $ \mathbb{Z}_2 $ polynomial ring, demonstrating how polynomials can be broken down into irreducible components within this specific setting. Understanding $ \mathbb{Z}_2[x] $ allows one to engrain themselves deeply into algebraic structures, especially those applicable in modern technologies like error-correcting codes.

91影视

Factor the indicated polynomial \(f(x)\) completely into irreducible factors in the polynomial ring \(F[x]\) for the indicated field \(F\). $$ f(x)=x^{4}+1 \quad F=\mathbb{Z}_{2} $$

Short Answer

Step by step solution

Understanding the Field and Polynomial

Expressing the Polynomial

Attempting Simple Factorization

Expanding and Matching

Solving the Equations

Choosing Possible Values

Final Factorization

Key Concepts

finite fields

irreducible polynomials

modular arithmetic

Z2 polynomial ring

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.