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Chapter 7: Problem 8

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ \mathrm{z}_{5} $$

Short Answer

Expert verified
The field of quotients of \( \mathbb{Z}_5 \) is \( \mathbb{Z}_5 \) itself, as it is already a field.

Step by step solution

01

Identify the Given Ring

The given ring is \( \mathbb{Z}_5 \), which denotes the set of integers modulo 5. This set consists of the elements \( \{0, 1, 2, 3, 4\} \), where addition and multiplication are defined under modulo 5 arithmetic.
02

Verify Conditions for a Field of Quotients

A field of quotients exists for a commutative ring with no zero divisors. Each non-zero element must have a multiplicative inverse in order for the ring to be integrally closed.
03

Confirm the Existence of Inverses for Nonzero Elements

In \( \mathbb{Z}_5 \), each non-zero element (1, 2, 3, 4) has a multiplicative inverse. For example, the inverse of 2 is 3, because \(2 \times 3 \equiv 1 \mod 5\). Similarly, other elements can be paired with their inverses, confirming that all non-zero elements have inverses.
04

Conclude the Existence of a Field of Quotients

Since \( \mathbb{Z}_5 \) is a finite field where every non-zero element has an inverse, it is already a field. A field of quotients is unnecessary because \( \mathbb{Z}_5 \) is self-contained as a field.

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Key Concepts

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

Modulo Arithmetic
Modulo arithmetic is a system of arithmetic for integers, where numbers wrap around after a certain value, known as the modulus. In our original exercise, the ring is \( \mathbb{Z}_5 \), which means we are working with integers modulo 5. This small set consists of numbers: 0, 1, 2, 3, and 4. Here's how modulo arithmetic affects operations:

  • **Addition & Subtraction:** Add or subtract numbers normally, then divide by the modulus. The remainder after division is the result. For example, in \( \mathbb{Z}_5 \), \( 3 + 4 = 7 \equiv 2 \) (mod 5) because 7 divided by 5 gives a remainder of 2.
  • **Multiplication:** Multiply the numbers as usual, but like addition, use the remainder after division by the modulus. For instance, \( 2 \times 3 = 6 \equiv 1 \) (mod 5).
Modulo arithmetic is crucial when examining rings like \( \mathbb{Z}_5 \), ensuring operations stay within the set of allowable numbers, preventing any overflow beyond the modulus.
Commutative Ring
A commutative ring is a set combined with two operations: addition and multiplication, that behaves similarly to how we traditionally understand these operations.

Key properties of a commutative ring include:

  • **Addition:** Commutative (\( a + b = b + a \)) and associative (\( (a + b) + c = a + (b + c) \)), with an additive identity (usually 0) and additive inverses.
  • **Multiplication:** Commutative (\( a \times b = b \times a \)) and associative (\( (a \times b) \times c = a \times (b \times c) \)), having a multiplicative identity (usually 1).
  • **Distributive Property:** Multiplication distributes over addition (\( a \times (b + c) = a \times b + a \times c \)).
For \( \mathbb{Z}_5 \), all these conditions are satisfied, considering elements can be combined in multiple ways, while results remain consistent thanks to the properties of modular arithmetic. This makes it a commutative ring, providing a robust framework for calculations.
Multiplicative Inverse
The multiplicative inverse of a number in a set is something that can be multiplied by the original number to yield the multiplicative identity, generally 1. In the realm of modular arithmetic, particularly within a set like \( \mathbb{Z}_5 \), each non-zero number has a multiplicative inverse.

For example:

  • 2 has an inverse of 3 because \( 2 \times 3 \equiv 1 \) (mod 5).
  • 3 has an inverse of 2, hence \( 3 \times 2 \equiv 1 \) (mod 5).
  • 4 has an inverse of 4 itself, as \( 4 \times 4 \equiv 1 \) (mod 5).
The significance of this within \( \mathbb{Z}_5 \) is that the existence of inverses for every non-zero element ensures it forms a field. This is because, in any field, every non-zero element must have an inverse, maintaining symmetry in operations.
Finite Field
A finite field, also known as a Galois field, is a field with a finite number of elements. The structure of finite fields is heavily defined by the modular arithmetic system they rely on. The prime example given in our exercise is \( \mathbb{Z}_5 \).

For \( \mathbb{Z}_5 \):

  • **Closure:** Both addition and multiplication remain within the set. Each operation on any pair of elements outputs a number within \( \{0, 1, 2, 3, 4\} \).
  • **Inverse Compliance:** Every non-zero element of \( \mathbb{Z}_5 \) has a multiplicative inverse, characteristic of field structures.
  • **Non-Zero Characteristic:** The field characteristic is 5, as it requires adding the multiplicative identity (1) to itself five times to return to the additive identity (0): \( 1 + 1 + 1 + 1 + 1 = 0 \) (mod 5).
In conclusion, \( \mathbb{Z}_5 \) stands as a finite field because it adheres to all field axioms with a limited, distinct set of elements, ensuring operations are predictable, repeatable, and contained within the set.

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Most popular questions from this chapter

In Exercises 1 through 10 determine whether the indicated set \(l\) is an ideal in the indicated ring \(R\). $$ I=\left\\{(2 x, 3 x) \mid x \in \mathbb{Z}_{6}\right\\} \text { in } R=\mathbb{Z}_{6} \times \mathbb{Z}_{6} $$

Show that if \(D\) is a finite integral domain, then \(|D|=p^{n}\) for some integer \(n>0\), where \(p=\operatorname{char} D\).

Let \(R\) be a commutative ring and \(I\) an ideal in \(R\). The radical of \(I\) is defined as follows: $$ \operatorname{rad}(I)=\left\\{a \in R \mid a^{n} \in I \text { for some } n \in \mathbb{Z}\right\\} $$ Show that \(\operatorname{rad}(I)\) is an ideal in \(R\) containing \(I\).

Let \(I=n \mathbb{Z}\) and \(J=m \mathbb{Z}\) be two ideals in \(\mathbb{Z}\). By the previous exercise, \(I+J\) is an ideal in \(\mathbb{Z}\) and therefore \(I+J=k \mathbb{Z}\) for some \(k\). Express \(k\) in terms of \(n\) and \(m\).

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ Q(t)=\left\\{a+b i \mid a, b \in Q, i^{2}=-1\right\\} $$
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