Constructing Addition and Multiplication Tables for a Field with 4 Elements (F₄)

Math Problem Statement

Inspired by his beloved TA, student X was curious whether he can construct a field with 4 elements. He defined a set with 4 elements by F₄ = {0, 1, 2, 3}. Write addition and multiplication tables for which the triplet (F₄, +, ⋅) is a field. Note: you may assume that such a field exists. You do not have to prove your statements.

Solution

To construct a field $\mathbb{F}_4$ with four elements, let's represent the elements of this field as $\{0, 1, a, a+1\}$ , where $a$ is a root of the irreducible polynomial $x^2 + x + 1 = 0$ over $\mathbb{F}_2$ (binary field). In this case, we have the following properties:

$a^2 = a + 1$ , because $a$ is defined as a root of $x^2 + x + 1 = 0$ .
The field operations need to satisfy all field axioms: closure, associativity, distributivity, identity elements, and inverses.

With this setup, here are the addition and multiplication tables for $\mathbb{F}_4$ .

Addition Table for $\mathbb{F}_4$ :

+ & 0 & 1 & a & a+1 \\ \hline 0 & 0 & 1 & a & a+1 \\ 1 & 1 & 0 & a+1 & a \\ a & a & a+1 & 0 & 1 \\ a+1 & a+1 & a & 1 & 0 \\ \end{array}$$ ### Multiplication Table for $$\mathbb{F}_4$$: $$\begin{array}{c|cccc} \cdot & 0 & 1 & a & a+1 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & a & a+1 \\ a & 0 & a & a+1 & 1 \\ a+1 & 0 & a+1 & 1 & a \\ \end{array}$$ These tables satisfy the field properties, including having an additive and a multiplicative identity, and every non-zero element has a multiplicative inverse. Would you like further clarification on constructing fields or any specific property? Here are some related questions to expand your understanding: 1. What is the significance of an irreducible polynomial in constructing finite fields? 2. Why does the field $$\mathbb{F}_4$$ only have four elements? 3. How are additive and multiplicative inverses determined in finite fields? 4. What are the properties that define a field in abstract algebra? 5. How do we verify that our tables satisfy field properties? **Tip:** In constructing fields, the choice of an irreducible polynomial is essential, as it ensures that elements are uniquely represented and the field properties are satisfied.

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Math Problem Analysis

Mathematical Concepts

Abstract Algebra
Finite Fields
Field Theory

Formulas

Addition and multiplication tables for a finite field

Theorems

Finite Field Theorem
Properties of Fields

Suitable Grade Level

Undergraduate Level

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