Problem 4 - A Course in Galois Theory

Question

Suppose that \(\varepsilon\) is a primitive \(m\) th root of unity over \(\mathbb{Q}\), where \(m>2\). Let \(\eta=\varepsilon+\varepsilon^{-1}\). Show that \([Q(\varepsilon): \mathbb{Q}(\eta)]=2\), find the minimal polynomial for \(\varepsilon\) over \(\mathbb{Q}(\eta)\) and identify the Galois group \(\Gamma[\mathbb{Q}(\varepsilon): \mathbb{Q}(\eta)]\)

Step-by-Step Solution

Answer

The degree \([\mathbb{Q}(\varepsilon): \mathbb{Q}(\eta)]\) is 2. The minimal polynomial is \(x^2 - \eta x + 1\). The Galois group is \(\mathbb{Z}_2\).

Step 1: Understand Primitive Root of Unity

A primitive \( m \)th root of unity, denoted \( \varepsilon \), is a complex number such that \( \varepsilon^m = 1 \) and \( \varepsilon^k eq 1 \) for any smaller positive integer \( k \). This ensures that \( \varepsilon\) is not just any root of unity but specifically one with minimal order \( m \).

Step 2: Define \( \eta \)

Given \( \varepsilon \) as a primitive \( m \)th root of unity, define \( \eta = \varepsilon + \varepsilon^{-1} \). Since \( \varepsilon^{-1} = \overline{\varepsilon} \) in this context, \( \eta \) is a real number. It's a trace element, representing the sum of \( \varepsilon \) and its complex conjugate.

Step 3: Determine a Minimal Polynomial for \( \varepsilon \) over \( \mathbb{Q}(\eta) \)

Consider the expression \( (x - \varepsilon)(x - \varepsilon^{-1}) = x^2 - \eta x + 1 \). Replace \( \varepsilon \) and \( \varepsilon^{-1} \) with any complex conjugate pair that satisfies this relation, confirming that the minimal polynomial for \( \varepsilon \) over \( \mathbb{Q}(\eta) \) is \( x^2 - \eta x + 1 \).

Step 4: Show the Degree is 2

The minimal polynomial \( x^2 - \eta x + 1 \) is of degree 2. This implies that the field extension degree \( [\mathbb{Q}(\varepsilon): \mathbb{Q}(\eta)] \) is indeed 2, as the degree of the polynomial is the minimal degree required to express \( \varepsilon \) over \( \mathbb{Q}(\eta) \).

Step 5: Identify the Galois Group

The Galois group \( \Gamma[\mathbb{Q}(\varepsilon): \mathbb{Q}(\eta)] \) corresponds to the symmetries in the roots of the polynomial \( x^2 - \eta x + 1 \). With two roots, \( \varepsilon \) and \( \varepsilon^{-1} \), the Galois group is isomorphic to \( \mathbb{Z}_2 \), the cyclic group of order 2, as it only swaps \( \varepsilon \) and \( \varepsilon^{-1} \).

Key Concepts

Primitive Roots of Unity

When we talk about primitive roots of unity, we are diving into the heart of complex numbers and their fascinating symmetries. A primitive root of unity is often denoted as \( \varepsilon \) and satisfies the condition \( \varepsilon^m = 1 \), where \( m \) is a positive integer. This means \( \varepsilon \) is a solution to the equation \( x^m - 1 = 0 \). But there's more! To be considered 'primitive', \( \varepsilon \) must not satisfy \( \varepsilon^k = 1 \) for any positive integer \( k < m \). Essentially, \( \varepsilon \) is the first root to complete the cycle after \( m \) applications.

• Primitive roots of unity are crucial for understanding symmetries in numbers.
• They are used extensively in fields like algebra and number theory.

An example could help visualize it: imagine the seismic cycle of a song. It repeats exactly every 10 seconds. If \( \varepsilon \) represents one full cycle, only at the 10-second mark does \( \varepsilon \) complete full circle and repeat. That's what a primitive root of unity does in terms of complex numbers.

Minimal Polynomial

The concept of a minimal polynomial is essential in field theory and is closely tied to the idea of algebraic numbers. When you have a complex number like \( \varepsilon \), which is a primitive \( m \)-th root of unity, its behavior can be expressed through a polynomial with the smallest possible degree.Such a polynomial for \( \varepsilon \) becomes vital as it provides the simplest way to describe \( \varepsilon \) over a given field, here \( \mathbb{Q}(\eta) \). Notably, for the expression \( \varepsilon + \varepsilon^{-1} \) as \( \eta \), the minimal polynomial turns out to be \( x^2 - \eta x + 1 \).

• The minimal polynomial is always monic, meaning the leading coefficient is 1.
• It provides insights into the degree of extension when moving from one field to another.

Think of the minimal polynomial as the most 'efficient' polynomial capturing all details about \( \varepsilon \) with respect to the field. Simplifying our understanding and computations, it's as if you found the shortest route explaining how this eager root of unity relates to its field.

Field Extensions

Field extensions are central in algebra, particularly in Galois Theory. When we seek to understand \( \mathbb{Q}(\varepsilon) \), a field extension of \( \mathbb{Q}(\eta) \), what we're really looking at is how one field can build upon another.In this specific case, our base field \( \mathbb{Q}(\eta) \) is extended to \( \mathbb{Q}(\varepsilon) \) using \( \varepsilon \), the primitive \( m \)-th root of unity. This is characterized by the expression \( [\mathbb{Q}(\varepsilon) : \mathbb{Q}(\eta)] = 2 \), indicating that the degree of the extension is 2.

• Field extensions illuminate how different field structures relate and interact.
• The degree of the extension indicates the 'dimensions' one field adds upon another.

Picture field extensions as building blocks: from a sturdy foundation, you add layers—each layer contributing new properties, expanding the mathematical 'universe' in novel ways.

Galois Groups

Galois groups are intriguing entities that provide a bridge between algebra and geometry by describing how roots of polynomials can be transformed. In the context of our primitive root of unity \( \varepsilon \), the Galois group \( \Gamma[\mathbb{Q}(\varepsilon) : \mathbb{Q}(\eta)] \) captures the symmetries in the polynomial \( x^2 - \eta x + 1 \).This polynomial has roots \( \varepsilon \) and \( \varepsilon^{-1} \), and the Galois group explains how these roots can 'swap' places. Structurally, it is isomorphic to the cyclic group \( \mathbb{Z}_2 \), indicating that there are only two elements, effectively swapping the roots.

• Galois groups tell us how fields can map onto each other.
• They offer a rich framework for understanding 'reversible' processes in algebra.

When thinking about Galois groups, imagine a mirror reflecting objects. The essence remains the same, but positions can swap—akin to how roots are exchanged through transformations described by the group.