Problem 9 - Abstract Algebra: Theory and Applications

Question

What is the characteristic of the field formed by the set of matrices $$ F=\left\\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right),\left(\begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right),\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right)\right\\} $$ with entries in \(\mathbb{Z}_{2} ?\)

Step-by-Step Solution

Answer

The set of matrices forms a field with the operations of addition and multiplication modulo 2. The characteristic of this field is 2 since the smallest positive integer n where n times the identity element equals the zero element is 2.

Step 1: Verify if the set forms a field

To verify if the given set of matrices forms a field, we need to check if the following field properties hold: 1. Closure under addition and multiplication. 2. Associativity of addition and multiplication. 3. Commutativity of addition and multiplication. 4. Existence of identity and inverse elements. Since all the matrix entries are in \(\mathbb{Z}_2\), the addition and multiplication of matrices will be modulo 2. Any operation between these matrices will still yield matrices with entries in \(\mathbb{Z}_2\). However, we should verify whether the result is also in set F. Checking closure for addition and multiplication, we find that all possible operations result in matrices contained in F. Thus, the set F is closed under addition and multiplication. Associativity and commutativity follow from the properties of matrix addition and multiplication with entries in \(\mathbb{Z}_2\). The identity matrix $\left(\begin{array}{ll} 1 & 0 \\\ 0 & 1 \end{array}\right)$ is already in the set F. Also, we can find the inverse matrices for the elements of the set F. Therefore, F satisfies all the properties of a field.

Step 2: Find the characteristic of the field

The characteristic of a field is the smallest positive integer n such that n times the identity element is equal to the zero element. In this case, the identity element is the identity matrix and the zero element is the zero matrix. We need to find the smallest positive integer n such that: $$ n\left(\begin{array}{ll} 1 & 0 \\\ 0 & 1 \end{array}\right) = \left(\begin{array}{ll} 0 & 0 \\\ 0 & 0 \end{array}\right) $$ Using matrix multiplication in \(\mathbb{Z}_2\), we find that: $$ 2\left(\begin{array}{ll} 1 & 0 \\\ 0 & 1 \end{array}\right) = \left(\begin{array}{ll} 0 & 0 \\\ 0 & 0 \end{array}\right) $$ So, the characteristic of the field formed by the set F is 2.

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental component of linear algebra. It involves operations like addition, subtraction, and multiplication applied to matrices. A matrix is a rectangular array of numbers arranged in rows and columns. In our scenario, we are dealing with matrices with 2 rows and 2 columns. Understanding matrix algebra requires knowledge of how these operations work:

• Addition: To add two matrices, add corresponding elements. This operation is straightforward and is performed element-wise.
• Multiplication: More complex than addition, matrix multiplication involves the dot product of rows and columns. For two matrices to multiply, the number of columns in the first must equal the number of rows in the second.
• Identity Matrix: Just like the number 1 in arithmetic, the identity matrix, often denoted as
  \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), is the multiplicative identity for matrices.

Matrix algebra extends these operations into practical applications, like solving sets of linear equations or transforming geometric objects.

Finite Fields

Finite fields, sometimes called Galois fields, are structures in algebra where there is a finite number of elements. In finite fields:

• All elements must obey the rules of both addition and multiplication, similar to ordinary arithmetic.
• Each finite field has a prime number of elements, denoted as \( \, \mathbb{Z}_p\, \), where the arithmetic operations are performed modulo \(\,p\,\).

In this exercise, our matrices have entries in \(\mathbb{Z}_2\), meaning they operate over a finite field with just two elements: 0 and 1. Each operation within this field loops back after reaching the field's characteristic, giving it a modular nature. Finite fields are crucial in areas like coding theory and cryptography because they provide a limited yet fully functional space of numbers.

Modular Arithmetic

Modular arithmetic is the system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, called the modulus. Think of it like the numbers on a clock.

• In modular arithmetic, numbers reset to zero once they reach a pre-defined number (the modulus).
• All calculations are performed with clock-like precision. For example, in \(\mathbb{Z}_2\), after 1, you revert back to 0 for addition \(\, (1 + 1 = 0)\, \).
• It's used extensively in computer science, particularly in algorithms and cryptography.

The concept is directly related to the exercise, where all matrix operations were conducted under \(\mathbb{Z}_2\). This means every matrix addition and multiplication was taken modulo 2, fitting within finite fields' and modular arithmetic's cyclical nature.