Problem 15 - Algebra: Pure and Applied

Question

In Exercises 14 through 17 find the order and characteristic of the indicated fields. $$ \mathbb{Z}[i] /\langle 7\rangle $$

Step-by-Step Solution

Answer

The field's order is 49, and its characteristic is 7.

Step 1: Understand the Field Structure

The expression \( \mathbb{Z}[i] / \langle 7 \rangle \) represents the quotient ring formed by dividing the Gaussian integers \( \mathbb{Z}[i] \) by the ideal generated by 7. The field has elements that are equivalence classes of Gaussian integers modulo the ideal \( \langle 7 \rangle \).

Step 2: Determine the Elements of the Field

The elements of \( \mathbb{Z}[i] / \langle 7 \rangle \) are equivalence classes of forms \( a + bi \) where \( a, b \) are integers ranging from 0 to 6, since we are considering equivalence modulo 7. This implies that there are \( 7 \times 7 = 49 \) possible elements.

Step 3: Finding the Order of the Field

Since the number of elements in the field is \( 49 \), the order of the field is 49.

Step 4: Determine the Characteristic of the Field

The characteristic of a finite field is the smallest positive integer \( n \) such that \( n \times 1 = 0 \) in the field. In \( \mathbb{Z}[i] / \langle 7 \rangle \), this is 7, as it comes directly from the ideal \( \langle 7 \rangle \), meaning performing addition 7 times of the multiplicative identity yields zero.

Key Concepts

Gaussian Integers

Gaussian Integers are a fascinating extension of the usual set of integers. They are represented in the form of \( a + bi \), where \( a \) and \( b \) are integers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \). This set, \( \mathbb{Z}[i] \), includes all numbers of this form and resembles a lattice of integer points in the complex plane.
- Just like regular integers, Gaussian integers can be added, subtracted, and multiplied.
- Division is also possible but not always yields another Gaussian integer, forming instead a ring.Understanding Gaussian integers is crucial when dealing with complex number fields, such as in this exercise where we look at the quotient ring \( \mathbb{Z}[i] / \langle 7 \rangle \).

Quotient Ring

A Quotient Ring \( \mathbb{Z}[i] / \langle 7 \rangle \) is formed when Gaussian integers are divided by an ideal, which in this case is generated by 7. This process creates a new set of elements, the equivalence classes.Each element in this quotient ring represents a group of Gaussian integers. Two integers are equivalent if their difference is divisible by 7.
- The notation \( \langle 7 \rangle \) signifies an ideal generated by 7. This notion helps to systematically work within modular arithmetic contexts.
- These equivalence classes form the elements of the quotient ring, thereby allowing us to explore properties such as field order and characteristic.

Finite Field

A finite field is an algebraic structure containing a finite number of elements. These fields are critical in number theory, cryptography, and coding theory. The finite field described by \( \mathbb{Z}[i] / \langle 7 \rangle \) contains 49 elements, made up of combinations \( a + bi \), where both \( a \) and \( b \) range from 0 to 6.- Finite fields, also called Galois fields, are defined fully by their characteristic and the number of elements (often called the order).
- Operations like addition and multiplication are performed modulo the ideal generating the field, in this instance \( 7 \). This property determines the size of the field.

Field Characteristic

The characteristic of a field is a fundamental concept in abstract algebra. It refers to the smallest positive integer \( n \) such that \( n \times 1 = 0 \) in the field.For the field \( \mathbb{Z}[i] / \langle 7 \rangle \), the characteristic is 7. This means that if you take the multiplicative identity (1) and add it to itself 7 times, the result is zero.
- In simple terms, the field "resets" after cycling through seven additions of 1.
- The characteristic helps in understanding how the field behaves under addition, and why it is defined in terms of the ideal \( \langle 7 \rangle \), from which the field derives its modular base.

Field Order

Field Order refers to the number of elements present in a field. For the quotient ring \( \mathbb{Z}[i] / \langle 7 \rangle \), this order is determined by counting all the unique equivalence classes that can be formed.- In this example, the order of the field is 49 because there are precisely 49 combinations of the elements \( a + bi \) when both \( a \) and \( b \) can independently take any of the 7 values from 0 to 6.
- Understanding the field order is essential for realizing the structure's size and properties, such as symmetry and the functioning of operations within it.