Problem 13 - A Course in Galois Theory

Question

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) \(\left\\{a+2^{1 / 3} b+4^{1 / 3} c: a, b, c \in \mathbb{Q}\right\\}\).

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \(\mathbb{C}\).

Step 3: 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.

Step 4: 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.

Step 5: 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.

Key Concepts

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.