Problem 3 - A Course in Galois Theory

Question

Let \(L_{0}=\mathbb{Q}, L_{1}=\mathbb{Q}\left(3^{1 / 2}\right), L_{2}=\mathbb{Q}\left(\left(3^{1 / 2}+1\right)^{1 / 2}\right)\). Show that \(L_{1}: L_{0}\) and \(L_{2}: L_{1}\) are both normal extensions but that \(L_{2}: L_{0}\) is not normal. Find the minimal polynomial of \(\left(3^{1 / 2}+1\right)^{1 / 2}\) over \(\mathbb{Q}\), and find its Galois group.

Step-by-Step Solution

Answer

Normal extensions: \(L_1: L_0\) and \(L_2: L_1\). Not normal: \(L_2:L_0\). Minimal polynomial: \(x^4 - 2x^2 - 2\). Galois group: \(C_2 \times C_2\).

Step 1: Understanding Normal Extensions

A field extension \(L:K\) is normal if it is the splitting field over \(K\) of some polynomial. To show that \(L_1: L_0\) is normal, note that \(L_1 = \mathbb{Q}(\sqrt{3})\), which is the splitting field over \(\mathbb{Q}\) of the polynomial \(x^2 - 3 = 0\). Hence, \(L_1: L_0\) is normal.

Step 2: Verify Normalcy of L2 Over L1

For \(L_2: L_1\), we can express \(L_2 = \mathbb{Q}(\sqrt{\sqrt{3} + 1})\). Consider the polynomial \(x^2 - (\sqrt{3} + 1) = 0\). This polynomial splits completely in \(L_2\), making \(L_2:L_1\) a normal extension.

Step 3: Check Normality of L2 Over L0

To check if \(L_2:L_0\) is normal, we need to see if all conjugates of \((\sqrt{3}+1)^{1/2}\) are contained within \(L_2\). The conjugates of \((\sqrt{3} + 1)\) are \(\pm\sqrt{3}+1\), and their square roots are not all in \(L_2\). Thus, \(L_2 : L_0\) is not normal.

Step 4: Find Minimal Polynomial

The minimal polynomial of \((\sqrt{3} + 1)^{1/2}\) over \(\mathbb{Q}\) is analyzed by setting \(y = (\sqrt{3} + 1)^{1/2}\) so that \(y^2 = \sqrt{3} + 1\) and \(y^4 - 2y^2 - 2 = 0\). Thus, the minimal polynomial is \(x^4 - 2x^2 - 2\).

Step 5: Determine the Galois Group

The Galois group of \(L_2\) over \(\mathbb{Q}\) is determined by investigating the roots of the minimal polynomial \(x^4 - 2x^2 - 2\). There are 4 roots, leading to potential permutations. The Galois group is \(C_2 \times C_2\), as each conjugate transformation results in a degree 2 part of the splitting field.

Key Concepts

Normal Extension

In Galois theory, understanding whether a field extension is normal is fundamental. A field extension \(L:K\) is called a normal extension if it is the splitting field of some polynomial over \(K\). This means that every root of the polynomial lies in \(L\).

For instance, consider \(L_1 = \mathbb{Q}(\sqrt{3})\). It is the smallest field encompassing \(\mathbb{Q}\) and \(\sqrt{3}\). The polynomial \(x^2 - 3 = 0\) has its roots, \(\sqrt{3}\) and \(-\sqrt{3}\), both within \(L_1\). Therefore, \(L_1 : L_0\) is normal.

On the other side, look at \(L_2 = \mathbb{Q}((\sqrt{3}+1)^{1/2})\). Dealing with the polynomial \(x^2 - (\sqrt{3}+1) = 0\), each root \( (-(\sqrt{3}+1))\) exists in \(L_2\) suggesting normalcy of \(L_2:L_1\). However, \(L_2: L_0\) requires all conjugates of \((\sqrt{3}+1)^{1/2}\), such as \((\sqrt{3}-1)^{1/2}\), to be present, but this isn't the case. Hence \(L_2:L_0\) is not normal.

Field Extension

Field extensions are crucial in understanding more complex algebraic structures. A field extension \(L:K\) occurs when the field \(L\) contains \(K\) as a subfield. This simply means that the elements of \(K\) are included within \(L\). These extensions help us analyze properties of polynomials and their roots.

In our scenario, \(L_1 = \mathbb{Q}(\sqrt{3})\) is a field extension over \(L_0 = \mathbb{Q}\). It involves introducing \(\sqrt{3}\), expanding the rational field to contain solutions to \(x^2 - 3 = 0\). Similarly, \(L_2 = \mathbb{Q}((\sqrt{3}+1)^{1/2})\) extends \(L_1\) by including \( (\sqrt{3}+1)^{1/2}\).

These extensions play a vital role in breaking down how algebraic manipulations can unfold within these larger fields. They offer a wider landscape to solve polynomial equations by introducing new elements not found in the base field.

Minimal Polynomial

A minimal polynomial is the lowest degree monic polynomial with coefficients in a given field that has the element as a root. It essentially provides the simplest polynomial relationship for a number. Finding the minimal polynomial is essential to analyzing the structure of field extensions.

To find the minimal polynomial of \((\sqrt{3} + 1)^{1/2}\) over \(\mathbb{Q}\), denote it as \(y\). We start with \(y^2 = \sqrt{3} + 1\). Rearranging gives us \(y^4 - 2y^2 - 2 = 0\), which cannot be reduced further over \(\mathbb{Q}\). Hence, \(x^4 - 2x^2 - 2\) is the minimal polynomial.

This polynomial tells us about the degree of the field extension \(L_2: L_0\), and highlights how the field \(L_2\) incorporates \((\sqrt{3} + 1)^{1/2}\) into its structure. It assists not only in solving equations in higher fields but also in understanding the properties that govern such extensions.

Galois Group

A Galois group is a group that represents the symmetries in the roots of a polynomial and helps in understanding how these roots can be interchanged while respecting the field's structure. It captures the essence of field automorphisms which keep the base field unchanged while manipulating the roots.

For \(L_2\) and its minimal polynomial \(x^4 - 2x^2 - 2\), the Galois group is quite enlightening. With four roots, permutations dictate the possible symmetries in \(\pm (\sqrt{3}+1)^{1/2}\) and their combinations.

The Galois group here is \(C_2 \times C_2\), representing the structure that maintains relationships among \(\pm (\sqrt{3}+1)^{1/2}\). This reflects two simple 2-cycles, indicating subtle yet powerful symmetries in polynomial solutions. Understanding this group is key to unravelling deeper algebraic equations and the behaviors of complex numbers in fields.