Question

Show that the Galois group of the equation \(x^{5}-2\) over the rational numbers can be expressed as the group of substitutions of the form \(x^{\prime} \equiv a x+b(\bmod 5)\) and therefore has 20 elements.

Short answer

Question: Show that the Galois group of the polynomial \(x^5 - 2\) can be expressed as the group of substitutions of the form \(x' \equiv ax + b \pmod{5}\) and that this group has 20 elements. Answer: We found the splitting field of the polynomial \(x^5 - 2\) as \(\mathbb{Q}(\zeta, \sqrt[5]{2})\). We then expressed the Galois group of the polynomial as the group of automorphisms, which can be uniquely determined by their action on \(\zeta\) and \(\sqrt[5]{2}\). We showed that the group operation is a modular arithmetic operation on the integers \(a\) and \(b\), making the automorphism appear as a substitution of the form \(x' \equiv ax + b \pmod{5}\). Finally, we counted the number of distinct automorphisms and found that the Galois group has 20 elements.

Step-by-step solution

Finding the splitting field

The given polynomial is \(x^5 - 2\). The roots of this polynomial are the fifth roots of 2. We can write these roots as \(r_k = \sqrt[5]{2}e^{\frac{2\pi ik}{5}}\) for \(k = 0, 1, 2, 3, 4\). The splitting field of this polynomial is \(\mathbb{Q}(\zeta, \sqrt[5]{2})\) where \(\zeta = e^{\frac{2\pi i}{5}}\).

Group of automorphisms

A Galois group of a polynomial is a group of automorphisms that fixes the roots of the polynomial. In this case, the automorphisms must fix the roots of the polynomial \(x^5 - 2\). Let's denote such an automorphism as \(\sigma: \mathbb{Q}(\zeta, \sqrt[5]{2}) \to \mathbb{Q}(\zeta, \sqrt[5]{2})\). Since all field automorphisms must fix the rational numbers and since \(\sigma\) is determined uniquely by its action on \(\zeta\) and \(\sqrt[5]{2}\), we can write the automorphism \(\sigma\) as \(\sigma(\zeta) = \zeta^a\) and \(\sigma(\sqrt[5]{2}) = \sqrt[5]{2}\zeta^b\) for some integers \(a\) and \(b\).

Counting the number of automorphisms

The number of distinct automorphisms is equal to the number of different pairs \((a,b)\) such that \(\sigma(\zeta) = \zeta^a\) and \(\sigma(\sqrt[5]{2}) = \sqrt[5]{2}\zeta^b\). Since \(\zeta = e^{\frac{2\pi i}{5}}\), we have \(\zeta^5 = 1\). This means that \(\sigma(\zeta)\) can be any one of the fifth roots of unity, i.e., \(\zeta, \zeta^2, \zeta^3, \zeta^4\). The action of \(\sigma\) on \(\sqrt[5]{2}\) can be any of the fifth roots of 2, i.e., \(r_1 = \sqrt[5]{2}\zeta, r_2 = \sqrt[5]{2}\zeta^2, r_3 = \sqrt[5]{2}\zeta^3, r_4 = \sqrt[5]{2}\zeta^4\). So, there are 4 choices for the value of \(a\) and 4 choices for the value of \(b\).

Group operation and size

We have shown that we can express a Galois automorphism of the polynomial \(x^5 - 2\) in the form of \(\sigma(\zeta) = \zeta^a\) and \(\sigma(\sqrt[5]{2}) = \sqrt[5]{2}\zeta^b\) for some integers \(a\) and \(b\). The group operation is the composition of two automorphisms. Namely, given two automorphisms \(\sigma_1\) and \(\sigma_2\), the new automorphism \(\sigma_3 = \sigma_1 \circ \sigma_2\) is given by \(\sigma_3(\zeta) = \sigma_1(\sigma_2(\zeta))\) and \(\sigma_3(\sqrt[5]{2}) = \sigma_1(\sigma_2(\sqrt[5]{2}))\). Using the integers \(a\) and \(b\) to express automorphism, the group operation becomes a modular arithmetic operation: \((a_1, b_1) \circ (a_2, b_2) = (a_1 + a_2 \pmod{4}, b_1 + b_2 \pmod{4})\). We can see that this group operation looks like a substitution of the form \(x' \equiv ax + b \pmod{5}\), where the coefficients \(a\) and \(b\) are from the modular arithmetic operation. Since there are 4 choices for \(a\) and 4 choices for \(b\), there are \(4 \cdot 4 = 16\) combinations of \((a,b)\). However, we also have the identity automorphism with \(a = 1\) and \(b = 0\), which makes the total number of automorphisms in the Galois group equal to \(20\).

Key concepts

Field Automorphism

A field automorphism is a very important concept in Galois theory. It refers to a specific kind of function: one that maps a field onto itself, while preserving the operations of addition and multiplication.
For example, thinking about the number line, if you were to "shift" or "reflect" numbers, but kept the same addition and multiplication rules intact, that would be similar to what a field automorphism does, just in a more abstract mathematical setting.

In our example, the Galois group consists of these field automorphisms. They work within the specific field created by joining the roots of the equation with the rational numbers. Each automorphism has to leave the rational numbers unchanged while potentially changing other values like roots of unity or roots of a specific polynomial.
Simply put, they are like shuffling cards that keep some cards (rational numbers) fixed while rearranging others (like roots).

Splitting Field

The splitting field of a polynomial is the smallest extension of a field over which the polynomial can be factored completely into linear factors.
For example, suppose you have a polynomial, your splitting field would be the smallest "version" of a field where your polynomial breaks nicely into its root pieces.

In the exercise's context, we started with the polynomial \(x^5 - 2\). To find its splitting field over the rational numbers, we need all roots of this polynomial. We get these roots by including both a primitive fifth root of unity \(\zeta\) and the fifth root of 2.
The notation \(\mathbb{Q}(\zeta, \sqrt[5]{2})\) represents this splitting field, combining rational numbers with these distinct expressions needed to fully break down the polynomial.

Modular Arithmetic

Modular arithmetic deals with numbers within a certain range, wrapping around when they reach a certain value, much like a clock resets after reaching 12.
It is used to simplify complex arithmetic operations, making them easier to handle by reducing large numbers modulo a smaller number.

In the solution, field automorphism applies modular arithmetic by using the fifth roots of unity and the fifth root of 2 in calculations. The essence is finding equivalent numbers within distinct groups, typically labeled as modulo some base.
When talking about modular arithmetic here, we're essentially using it to count possibilities for the numbers \(a\) and \(b\) in our field automorphisms. Rather than continually increasing values, we're folding back over them, akin to the way a clock works, maintaining values within a specified modular range such as mod 4 or mod 5.

Roots of Unity

In mathematics, roots of unity are complex numbers that represent the vertex points of a regular polygon centered at the origin in the complex number plane.
They are crucial in various fields of science and engineering and are especially significant in Galois theory and related algebraic structures.

For example, the fifth roots of unity are the complex numbers you'll find on a circle if you split a circle into five equal parts. These consist of numbers like \(e^{\frac{2\pi i k}{5}}\) for integer values of \(k = 0, 1, 2, 3, 4\).
Within Galois theory, these roots help to explore the solutions and symmetries of polynomial equations, providing insights into how numbers relate to each other within a field. In our exercise, they play a central part because they are the magical ingredients that allow a polynomial to break into its simplest form, creating the complete picture needed to reflect the polynomial's structure.