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Chapter 7: Problem 31

Write down the multiplication table for \(\mathbb{Z}_{2}[X] /\left(X^{2}+X\right)\). Is this a field?

Short Answer

Expert verified
Answer: Yes, the quotient ring \(\mathbb{Z}_{2}[X]/(X^2 + X)\) is a field, as every nonzero element has a multiplicative inverse.

Step by step solution

01

Identify the elements of the ring

The given ring is \(\mathbb{Z}_{2}[X]/(X^2 + X)\), which means we are working in the polynomial ring \(\mathbb{Z}_2[X]\) (polynomials with coefficients in \(\mathbb{Z}_2\)) modulo the ideal generated by \(X^2 + X\). We know that \(\mathbb{Z}_2\) contains only two elements: 0 and 1. So, let's consider the polynomials with degree less than 2 (since we are working modulo \(X^2 + X\)): 0, 1, X, and 1 + X These are the elements of the given ring.
02

Construct the multiplication table

Next, we find all the products among these elements, simplifying the results modulo the given ideal \((X^2 + X)\). Here is the multiplication table: ``` 0 1 X 1+X +-------------- 0 | 0 0 0 0 1 | 0 1 X 1+X X | 0 X 1+X 1 1+X | 0 1+X 1 X ```
03

Determine if it's a field

In order to determine if this is a field, we need to check if every nonzero element has a multiplicative inverse. Let's analyze every nonzero element: 1. The inverse of 1 is itself, since \(1 \cdot 1 \equiv 1 \pmod{X^2+X}\). 2. The inverse of X is X, as \(X \cdot X \equiv X^2 \equiv 1 \pmod{X^2+X}\). 3. The inverse of 1+X is also 1+X, because \((1+X) \cdot (1+X) \equiv 1 + 2X + X^2 \equiv 1+X^2 \equiv 1+(-X) \equiv 1+X \pmod{X^2+X}\). In all cases, we found a multiplicative inverse for every nonzero element. Therefore, the given ring \(\mathbb{Z}_2[X]/(X^2+X)\) is indeed a field.

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Key Concepts

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

Modular Arithmetic
Understanding modular arithmetic is essential for working with polynomial rings in advanced mathematics. It involves performing calculations with integers where numbers wrap around after reaching a certain value, known as the modulus. For example, clock arithmetic is a type of modular arithmetic with a modulus of 12, since after 12 hours, the count starts over from 1.

In the context of polynomial rings, modular arithmetic operates on polynomial coefficients under a specified modulus. For instance, in the ring \(\mathbb{Z}_{2}[X]\), the modulus is 2. This means we apply arithmetic operations on coefficients and then take the remainder when divided by 2. Here, any coefficient is either 0 or 1, because any even number modulo 2 equates to 0, and any odd number modulo 2 equates to 1.

This constraint simplifies operations within the polynomial ring, particularly the multiplication of polynomials. In our example, when we construct the multiplication table, we compute products of elements and express the results in terms of the ring's representatives, all under modulo 2 arithmetic. Understanding this process is integral to grasping subsequent concepts such as field theory and constructing multiplication tables for such polynomial rings.
Field Theory
Field theory is a branch of abstract algebra that deals with the study of fields — mathematical structures that allow addition, subtraction, multiplication, and division operations to be performed while satisfying certain axioms. A crucial aspect of a field is the existence of a multiplicative inverse for every nonzero element.

In simpler terms, if we look at a set and it allows us to add, subtract, multiply, and divide any two elements (except division by zero) in such a way that the rules we're familiar with for rational numbers hold, then we call this set a field. These rules include commutative, associative, and distributive properties, as well as the existence of additive and multiplicative identities and inverses.

Using the exercise as an example, the solution shows that the ring \(\mathbb{Z}_2[X]/(X^2 + X)\) is indeed a field. This conclusion is reached by showing that each nonzero element within the ring has a multiplicative inverse, thereby satisfying the necessary condition for a field. A deep understanding of field theory can reveal why certain mathematical structures, like our polynomial ring, can (or cannot) form a field and the implications it has in various areas of mathematics and science, such as coding theory and cryptography.
Multiplication Table
A multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. The table is a convenient tool to visualize how elements within a set multiply with one another, especially within finite groups, rings, and fields. In our polynomial ring example, constructing a multiplication table assists in verifying whether the algebraic structure fulfills the criteria of a field.

The process involves listing all elements of the ring along two axes of a grid and then filling in the grid with the product of the elements corresponding to the given row and column, simplifying as needed based on the ring's rules. In \(\mathbb{Z}_{2}[X]/(X^{2}+X)\), we consider polynomials of degree less than 2 and apply modulo 2 arithmetic to determine the outcomes. The multiplication table from the exercise demonstrates that the structure allows for basic algebraic operations to be performed, and every nonzero element pairs with an inverse.

Creating and analyzing such tables can provide immediate insight into the structure of the algebraic system. It is a powerful tool for both learning and teaching because it helps to make abstract concepts like polynomial ring multiplication more tangible.

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Most popular questions from this chapter

Consider the subring \(\mathbb{Z}[1 / 2]\) of \(\mathbb{Q} .\) Show that \(\mathbb{Z}[1 / 2]=\left\\{a / 2^{i}:\right.\) \(a, i \in \mathbb{Z}, i \geq 0\\},\) that \((\mathbb{Z}[1 / 2])^{*}=\left\\{2^{i}: i \in \mathbb{Z}\right\\},\) and that every non-zero ideal of \(\mathbb{Z}[1 / 2]\) is of the form \((m),\) for some uniquely determined, odd integer \(m .\)

Let \(R\) be a ring, let \(g \in R[X],\) with \(\operatorname{deg}(g)=k \geq 0,\) and let \(x\) be an element of \(R\). Show that: (a) there exist an integer \(m,\) with \(0 \leq m \leq k,\) and a polynomial \(q \in R[X],\) such that $$ g=(X-x)^{m} q \text { and } q(x) \neq 0 $$ and moreover, the values of \(m\) and \(q\) are uniquely determined; (b) if we evaluate \(g\) at \(X+x,\) we have $$ g(X+x)=\sum_{i=0}^{k} b_{i} X^{i} $$ where \(b_{0}=\cdots=b_{m-1}=0\) and \(b_{m}=q(x) \neq 0\)

It is a fact that every non-trivial ring \(R\) contain at least one maximal ideal. Showing this in general requires some fancy set-theoretic notions. This exercise develops a simple proof in the case where \(R\) is countable (see \(\S\) A3). (a) Show that if \(R\) is non-trivial but finite, then it contains a maximal ideal. (b) Assume that \(R\) is countably infinite, and let \(a_{1}, a_{2}, a_{3}, \ldots\) be an enumeration of the elements of \(R\). Define a sequence of ideals \(I_{0}, I_{1}, I_{2}, \ldots,\) as follows. Set \(I_{0}:=\left\\{0_{R}\right\\},\) and for each \(i \geq 0,\) define $$ I_{i+1}:=\left\\{\begin{array}{ll} I_{i}+a_{i} R & \text { if } I_{i}+a_{i} R \subsetneq R \\ I_{i} & \text { otherwise } \end{array}\right. $$ Finally, set \(I:=\bigcup_{i=0}^{\infty} I_{i},\) which by Exercise 7.37 is an ideal of \(R\). Show that \(I\) is a maximal ideal of \(R\). Hint: first, show that \(I \subsetneq R\) by assuming that \(1_{R} \in I\) and deriving a contradiction; then, show that \(I\) is maximal by assuming that for some \(i=1,2, \ldots,\) we have \(I \subsetneq I+a_{i} R \subsetneq R,\) and deriving a contradiction.

Let \(a, b\) be elements of a ring \(R\). Show that $$ a \mid b \Longleftrightarrow b \in a R \Leftrightarrow b R \subseteq a R $$

Let \(F\) be a field. (a) Show that for all \(b \in F,\) we have \(b^{2}=1\) if and only if \(b=\pm 1\). (b) Show that for all \(a, b \in F,\) we have \(a^{2}=b^{2}\) if and only if \(a=\pm b\). (c) Show that the familiar quadratic formula holds for \(F\), assuming \(F\) has characteristic other than 2 , so that \(2_{F} \neq 0_{F}\). That is, for all \(a, b, c \in F\) with \(a \neq 0,\) the polynomial \(g:=a X^{2}+b X+c \in F[X]\) has a root in \(F\) if and only if there exists \(e \in F\) such that \(e^{2}=d,\) where \(d\) is the discriminant of \(g,\) defined as \(d:=b^{2}-4 a c,\) and in this case the roots of \(g\) are \((-b \pm e) / 2 a\)
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