Question

Let \(K\) be a vector space over \(\mathbb{Z}_{2}\) with basis \(\\{1, t\\},\) so \(K=\left\\{a+b t \mid a, b,\right.\) in \(\left.\mathbb{Z}_{2}\right\\} .\) It is known that \(K\) becomes a field of four elements if we define \(t^{2}=1+t\) Write down the multiplication table of \(K\).

Short answer

The multiplication table considers the elements 0, 1, \( t \), and \( 1+t \), adhering to \( t^2 = 1+t \).

Step-by-step solution

Understanding Vector Space Over \( \mathbb{Z}_2 \)

In this exercise, we are working within a vector space over the finite field \( \mathbb{Z}_2 \), which contains two elements, 0 and 1. The elements of this vector space are of the form \(a + bt\), where \(a\) and \(b\) are in \(\mathbb{Z}_2\). This means \(a\) and \(b\) can be either 0 or 1.

Basis Elements of K

Given the basis \( \{ 1, t \} \), the elements in \( K \) consist of all possible linear combinations of 1 and \( t\), using coefficients from \( \mathbb{Z}_2 \). This gives us four possible elements: \( 0 \), \( 1 \), \( t \), and \( 1+t \).

Field Definition and Relation

The problem states that \( K \) becomes a field when defining the relation \( t^2 = 1 + t \). This relation will guide us in establishing the multiplication rules among the elements of \( K \).

Formulating the Multiplication Table

To create the multiplication table for \( K \), calculate the products for each combination of the basis elements. Remember to use \( t^2 = t + 1 \) for substitution when necessary. Thus, calculate:- \(1 imes 1 = 1\)- \(1 imes t = t \)- \(1 imes (1 + t) = 1 + t \)- \(t imes t = 1 + t \) (using \(t^2 = 1 + t\))- \(t imes (1 + t) = t + t^2 = t + (1 + t) = 1\)- \((1 + t) imes (1 + t) = t^2 + 2t + 1 = (1 + t) + 0 + 1 = t\)

Constructing the Multiplication Table

With the calculated products, construct the multiplication table for the four elements 0, 1, \( t \), and \( 1+t \): \[ \begin{array}{c|cccc} \cdot & 0 & 1 & t & 1+t \ \hline 0 & 0 & 0 & 0 & 0 \ 1 & 0 & 1 & t & 1+t \ t & 0 & t & 1+t & 1 \ 1+t & 0 & 1+t & 1 & t \ \end{array} \]

Key concepts

Vector Spaces

A vector space is a fundamental concept in linear algebra. In simple terms, it's a collection of objects called vectors, which can be added together and multiplied by numbers (scalars) to produce another vector. Here, we are considering the vector space over the field \( \mathbb{Z}_2 \), which is the set \( \{0, 1\} \) with binary addition and multiplication.

In this context, the vector space \( K \) is formed by linear combinations of basis elements using coefficients from \( \mathbb{Z}_2 \). The basis elements here are \( 1 \) and \( t \). Only using 0 and 1 as coefficients limits our calculations, making it an ideal exploration space to comprehend vector space operations in finite fields. Vector spaces have essential properties, including closure under addition and scalar multiplication, the existence of a zero vector, and the ability to each multiply vector by scalars from a fixed field.

Basis Elements

The concept of basis elements is pivotal in understanding vector spaces. A basis is a set of vectors in a vector space such that every vector in the space can be expressed as a unique combination of the basis vectors. In our example, the basis \( \{ 1, t \} \) is used.

The elements in our vector space \( K \) are linear combinations of 1 and \( t \). With coefficients from \( \mathbb{Z}_2 \), the only available elements are 0, 1, \( t \), and \( 1+t \).

• 0: the identity element for addition
• 1: the multiplicative identity
• \( t \): a basis element that allows for expansion of the space
• \( 1+t \): another vector formed by adding 1 and \( t \)

Each element plays a unique role in forming the complete vector space, allowing all vectors within the space to be expressed as linear combinations of the basis.

Field Theory

Field theory is the study of algebraic structures known as fields. Fields are essentially sets equipped with two operations, addition and multiplication, that satisfy certain properties. These properties include the existence of additive and multiplicative identities, additive and multiplicative inverses for every element (except for the additive identity in the case of multiplicative inverses), and the operations of addition and multiplication being both commutative and associative.

In the exercise, \( K \) becomes a field when we introduce the relation \( t^2 = 1 + t \). This transformation defines rules that allow \( K \) to fulfill all the field properties. Constructing a field out of the vector space \( K \) involves creating a multiplication table that respects this relational structure. As a result of this field definition, we must consider how each multiplication operation influences the elements in terms of our new field theory boundary.

Discrete Mathematics

Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. This contrasts with continuous mathematics, which deals with objects that can vary smoothly.

The concept of finite fields, such as \( \mathbb{Z}_2 \), falls squarely within discrete mathematics. Finite fields are crucial in many applications, including coding theory and cryptography, because they provide a clean, finite environment to conduct arithmetic.

• Apply finite fields in computations involving discrete structures.
• Use them to understand vector spaces over these fields, like \( K \).
• Facilitate operations in modular environments, which underpin numerous algorithms and systems in computer science.

Understanding integer arithmetic and properties in finite settings enables us to solve many real-world problems that arise in digital computation and data transmission.