/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Which of the following subsets o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 13

Which of the following subsets of \(\mathbb{C}\) are subfields of \(C\) ? (i) \(\\{a+i b: a, b \in \mathbb{Q}\\}\). (ii) \(\left\\{a+\omega b: a, b \in \mathbb{Q}, \omega=\frac{1}{2}(-1+\sqrt{3} \mathrm{i})\right\\}\) (iii) \(\left\\{a+2^{1 / 3} b: a, b \in \mathbb{Q}\right\\}\). (iv) \(\left\\{a+2^{1 / 3} b+4^{1 / 3} c: a, b, c \in \mathbb{Q}\right\\}\).

Short Answer

Expert verified
Subsets (i) and (ii) are subfields of \(\mathbb{C}\); (iii) and (iv) are not.

Step by step solution

01

Define the Criteria for a Subfield

A subset of \(\mathbb{C}\) is a subfield if it is closed under addition, subtraction, multiplication, and division (except by zero). That means if \(x\) and \(y\) are in the subset, then \(x+y\), \(x-y\), \(xy\), and \(\frac{x}{y}\) (\(y eq 0\)) must also be in the subset.
02

Check Subset (i)

Consider the set \(\{a+ib : a, b \in \mathbb{Q}\}\). This is equivalent to \(\mathbb{Q}[i]\), the set of complex numbers with rational real and imaginary parts. \(\mathbb{Q}[i]\) forms a field as it is closed under addition, subtraction, multiplication, and division. Thus, subset (i) is a subfield of \(\mathbb{C}\).
03

Check Subset (ii)

For the set \(\{a+\omega b : a, b \in \mathbb{Q}, \omega = \frac{1}{2}(-1+\sqrt{3}i)\}\), \(\omega\) is a primitive cube root of unity. This set is \(\mathbb{Q}[\omega]\), which forms a field, known as the cyclotomic field of third roots of unity. Therefore, subset (ii) is a subfield.
04

Check Subset (iii)

Examine the set \(\{a+2^{1/3} b : a, b \in \mathbb{Q}\}\). Checking if it is closed under division is key. If we consider \(\frac{1}{2^{1/3}} = 2^{-1/3}\), this number might not be expressible as \(a+2^{1/3} b\) with rational coefficients. Thus, (iii) is not generally closed under division, and therefore not a subfield.
05

Check Subset (iv)

Look at the set \(\{a + 2^{1/3} b + 4^{1/3} c : a, b, c \in \mathbb{Q}\}\). Consider elements like \(2^{1/3}\) and \(4^{1/3}\). Division within this set might lead to elements not of the given form, due to irrational numbers involved. As closure under division could fail, (iv) is not a subfield.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Complex Numbers
Complex numbers are an extension of the real numbers and are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\). This mathematical concept allows us to work with numbers that extend beyond the real number line. Complex numbers can be represented as points in a plane using their real and imaginary parts.
  • The horizontal axis represents the real part \(a\), and the vertical axis represents the imaginary part \(b\).
  • The modulus of a complex number is its "distance" from the origin in this plane, calculated as \(|a + bi| = \sqrt{a^2 + b^2}\).
  • The argument is the angle the line connecting the origin to the point makes with the positive real axis, often calculated using \(\tan^{-1}(b/a)\).
Complex numbers are crucial in solving equations that do not have real solutions, like \(x^2 + 1 = 0\). They have applications across various domains like engineering, physics, and computer science.
Subfields
In the realm of field theory, a subfield is essentially a smaller field contained within a larger field, in this case, the field of complex numbers \(\mathbb{C}\). For a subset of \(\mathbb{C}\) to qualify as a subfield, it needs to meet certain criteria:
  • Closure under addition: If \(x\) and \(y\) are in the subset, \(x + y\) must also be in the subset.
  • Closure under subtraction: If \(x\) and \(y\) are in the subset, \(x - y\) must also be in the subset.
  • Closure under multiplication: If \(x\) and \(y\) are in the subset, \(x \cdot y\) must also be in the subset.
  • Closure under division: If \(x\) and \(y\) are in the subset and \(y eq 0\), \(\frac{x}{y}\) must also be in the subset.
Subfields retain the properties of a field, enabling deeper study and application of algebraic structures in mathematical theories. Examples include rational numbers, real numbers, and certain specialized fields like \(\mathbb{Q}[i]\) or cyclotomic fields.
Field Theory
Field theory is a branch of mathematics that studies the structure of fields, which are algebraic structures where you can perform addition, subtraction, multiplication, and division (by non-zero elements) operations. Here are some key points:
  • A field must satisfy several axioms, like associativity, commutativity for addition and multiplication, the existence of additive and multiplicative identities, and the existence of inverses for every non-zero element.
  • Fields are fundamental in many areas of mathematics, including algebraic geometry, number theory, and cryptography.
  • They allow for the construction of more complex number systems and provide a framework for understanding polynomials, equations, and algebraic structures.
Field theory is particularly significant in understanding Galois theory, which relates field extensions to group theory. This connection helps solve polynomial equations and analyze their solvability by radicals.
Rational Numbers
Rational numbers are numbers that can be expressed as the quotient of two integers, \(a/b\), where \(b eq 0\). They include integers, fractions, and terminating or repeating decimals. Here are some basic properties:
  • Rational numbers are denoted by \(\mathbb{Q}\).
  • They are a subset of real numbers and can therefore be plotted on a continuous number line.
  • Rational numbers are closed under addition, subtraction, multiplication, and division (provided we don't divide by zero).
In the context of field theory, rational numbers form a field because they satisfy the properties required of a field: closure, associativity, commutativity, distributivity, and the presence of inverses. Rational numbers serve as the building blocks for more complex number fields, and they are essential in constructing other mathematical entities like polynomials or rational functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that \(H\) is a normal subgroup of \(G\) and that \(K\) is a normal subgroup of \(H\). Is \(K\) necessarily a normal subgroup of \(G ?\)

Describe the elements of \(\mathbb{Z}_{n}\) which generate \(\mathbb{Z}_{n}\) (for any positive integer \(n\) ).

Every field has at least two elements. Show that there is a field with exactly two elements.

\(\mathbb{R}\) can be considered as a vector space over \(\mathbb{Q}\). Show that \(\mathbb{R}\) is not finite dimensional over Q. Can you find an infinite subset of \(R\) which is linearly independent over \(Q\) ?

If \(\pi \in \Sigma_{n}\), let \(\varepsilon(\pi)=\prod_{i
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.