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Magazine Chapter 8: Problem 5
Describe the indicated quotient rings. $$ \mathbb{Z}_{2}[x] /\left\langle x^{3}+x^{2}+x\right\rangle $$
Short Answer
Expert verified
The quotient ring is isomorphic to the set of polynomials of degree less than 3, formed by \(1, x, x^2\) with \(\mathbb{Z}_2\) coefficients.
Step by step solution
01
Identify the base ring and ideal
The ring is a polynomial ring over the field \(\mathbb{Z}_2\), noted as \(\mathbb{Z}_2[x]\). The ideal \(\langle x^3 + x^2 + x \rangle\) is generated by the polynomial \(x^3 + x^2 + x\).
02
Simplify the generating polynomial
The polynomial \(x^3 + x^2 + x\) can be factored as \(x(x^2 + x + 1)\). Within \(\mathbb{Z}_2\), this polynomial can potentially give expressions for division in the quotient ring.
03
Define the elements of the quotient ring
In the quotient ring \(\mathbb{Z}_2[x]/\langle x^3 + x^2 + x \rangle\), elements are equivalence classes of polynomials in \(\mathbb{Z}_2[x]\) divided by the ideal. This means every element in the quotient ring is of the form \(a(x) + \langle x^3 + x^2 + x \rangle\), where the degree of \(a(x)\) is less than 3.
04
Describe the structure
The structure of this quotient ring can be represented by a set of polynomials \(a_2x^2 + a_1x + a_0\) with coefficients \(a_2, a_1, a_0 \in \mathbb{Z}_2\). The addition and multiplication follow the operations modulo \(x^3 + x^2 + x\).
05
Use the Simplified Basis
Because the polynomial can be reduced as \(x^2 + x + 1\), the polynomial \(x^3\) can be rewritten in terms of lower degree polynomials. Hence, elements can be reduced even further using repeated substitution from the generators, simplifying to a basis consisting of \(1, x, x^2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Rings
Polynomial rings are at the heart of algebra, giving us a way to work with polynomials within a given set of numbers. A polynomial ring is denoted as \( R[x] \), meaning all polynomials with coefficients from a ring \( R \). In the exercise, \( \mathbb{Z}_2[x] \) is a polynomial ring with coefficients in \( \mathbb{Z}_2 \), which means each coefficient is either 0 or 1.
- Each polynomial is a sum of terms consisting of coefficients from \( \mathbb{Z}_2 \) multiplied by powers of \( x \).
- For example, \( 1 + x \) and \( x^2 + x + 1 \) are polynomials in \( \mathbb{Z}_2[x] \).
- The operations of addition and multiplication in polynomial rings work similarly to those over integers but follow modulo 2 arithmetic.
Ideal Theory
In algebra, ideals help us understand the structure of rings more deeply. An ideal of a ring is a subset that is closed under addition and also absorbs multiplication by any element in the ring. In the context of polynomial rings, ideals are often generated by polynomials.
- Given a polynomial \( f(x) \), the smallest ideal containing \( f(x) \) includes all multiples of \( f(x) \) by other polynomials.
- In \( \mathbb{Z}_2[x] \), the ideal \( \langle x^3 + x^2 + x \rangle \) is generated by the polynomial \( x^3 + x^2 + x \).
- All polynomials in this ideal can be expressed as \( g(x)(x^3 + x^2 + x) \) for some polynomial \( g(x) \).
Field of Two Elements
The field of two elements, \( \mathbb{Z}_2 \), consists of just two numbers: 0 and 1. This field obeys the rules of arithmetic familiar from school, but all operations are taken modulo 2, meaning results are reduced to 0 or 1.
- Addition and multiplication are defined such that: \( 1 + 1 = 0 \), \( 1 + 0 = 1 \), and multiplication follows the regular pattern \( 1\times 1 = 1 \).
- Thus, every polynomial in \( \mathbb{Z}_2[x] \) is composed of terms with coefficients that are either 0 or 1.
- This simplicity gives \( \mathbb{Z}_2 \) a unique and advantageous role in polynomial theory, especially when forming quotient rings.
Equivalence Classes
Equivalence classes group elements of a set together under an equivalence relation, expressing them as equivalent in some meaningful way.
In quotient rings, equivalence classes form by partitioning a polynomial ring by an ideal, essentially grouping together all polynomials that differ by a multiple of a given polynomial.
In quotient rings, equivalence classes form by partitioning a polynomial ring by an ideal, essentially grouping together all polynomials that differ by a multiple of a given polynomial.
- In \( \mathbb{Z}_2[x]/\langle x^3 + x^2 + x \rangle \), each equivalence class contains all polynomials that become identical when reduced by \( x^3 + x^2 + x \).
- This reduction process results in a simpler representation, consisting of polynomials with degrees lower than the generating polynomial (in this case, degree less than 3).
- For instance, \( x^3 \equiv - x^2 - x \equiv x^2 + x \mod (x^3 + x^2 + x) \), so in the quotient ring, \( x^3 \) is in the class of \( x^2 + x \).
