Problem 13 - A Computational Introduction to Number Theory and Algebra

Question

Consider the real numbers \(\sqrt{2}\) and \(\sqrt{3}\). Show that \((\mathbb{Q}[\sqrt{2}, \sqrt{3}]: \mathbb{Q})=(\mathbb{Q}[\sqrt{2}+\sqrt{3}]: \mathbb{Q})=4\)

Step-by-Step Solution

Answer

Answer: Both \((\mathbb{Q}[\sqrt{2}, \sqrt{3}]: \mathbb{Q})\) and \((\mathbb{Q}[\sqrt{2}+\sqrt{3}]: \mathbb{Q})\) have a degree of 4.

Step 1: Understand the field extensions

In this problem, we have two field extensions to work with: \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\) and \(\mathbb{Q}[\sqrt{2}+\sqrt{3}]\). The first extension contains all numbers that can be formed by combining rational numbers with \(\sqrt{2}\) and \(\sqrt{3}\) through arithmetical operations. The second extension contains all numbers that can be formed by combining rational numbers with the single number \(\sqrt{2}+\sqrt{3}\) through the same operations. We need to find the basis for both of these fields over \(\mathbb{Q}\). -Step 2: Determine the basis for \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\)

Step 2: Determine the basis for \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\)

Consider the numbers \(1, \sqrt{2}, \sqrt{3}\), and \(\sqrt{6}\) in the extension \(E = \mathbb{Q}[\sqrt{2}, \sqrt{3}]\). It is easy to see that these numbers are linearly independent over \(\mathbb{Q}\). Any linear combination of these numbers must have the form \(a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}\), where \(a,b,c,d \in \mathbb{Q}\). If we show that \(E\) is contained in the vector space generated by these 4 numbers, then we would know that the basis for \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\) has 4 elements. Observe that any element in \(E\) can be expressed in the form \((p_1 + q_1\sqrt{2}) + (p_2 + q_2\sqrt{2})\sqrt{3} = (p_1 + p_2\sqrt{3}) + (q_1 + q_2\sqrt{3})\sqrt{2}\) . Thus, any element in E can indeed be expressed as a linear combination of these 4 numbers. Therefore, the basis of \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\) is \(\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}\), and its degree is 4. -Step 3: Determine the basis for \(\mathbb{Q}[\sqrt{2}+\sqrt{3}]\)

Step 3: Determine the basis for \(\mathbb{Q}[\sqrt{2}+\sqrt{3}]\)

Let's now consider the extension \(F = \mathbb{Q}[\sqrt{2}+\sqrt{3}]\). We can express each of the elements in the basis of the previous extension, \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\), in terms of elements in \(F\). Observe that \((\sqrt{2} + \sqrt{3})^2 = 2 + 3 + 2\sqrt{6} = 5 + 2\sqrt{6}\) and \((\sqrt{2} + \sqrt{3})^3 = 11\sqrt{2} + 9\sqrt{3}\). Using these expressions, we can determine the relationships between the basis elements of \(E\) and elements in \(F\). For instance, \(\sqrt{2} = \frac{1}{9}((\sqrt{2} + \sqrt{3})^3 - 6(\sqrt{2} + \sqrt{3})^2 + 21(\sqrt{2} + \sqrt{3}))\). Similarly, we have \(\sqrt{3} = \frac{1}{3}(\sqrt{2} + \sqrt{3})(2(\sqrt{2} + \sqrt{3}) - (\sqrt{2} + \sqrt{3})^2)\). Since the elements \(1, \sqrt{2}, \sqrt{3}\), and \(\sqrt{6}\) can be expressed in terms of elements in \(F\), we know that the degree of \(F\) is at least 4. However, the degree of any simple extension of the form \(\mathbb{Q}[\alpha]\) cannot exceed the degree of the minimal polynomial of \(\alpha\). The minimal polynomial of \(\sqrt{2}+\sqrt{3}\) is \(x^4 - 10x^2 + 1\); thus, the degree of \(F\) is at most 4. Therefore, the degree of \(\mathbb{Q}[\sqrt{2}+\sqrt{3}]\) is also exactly 4. In conclusion, we have shown that both \((\mathbb{Q}[\sqrt{2}, \sqrt{3}]: \mathbb{Q})\) and \((\mathbb{Q}[\sqrt{2}+\sqrt{3}]: \mathbb{Q})\) are equal to 4, as required.

Key Concepts

Degree of Field Extension

When studying field extensions, it's crucial to understand the concept of the degree of a field extension. The degree of an extension, denoted \([L:K]\), is the dimension of the extension field \(L\) as a vector space over the base field \(K\). In other words, it refers to the number of elements in a basis that spans \(L\) over \(K\).

• If \(L\) can be expressed as \(K[\alpha]\), where \(\alpha\) is an element in \(L\), the degree of the extension is the degree of the minimal polynomial of \(\alpha\) over \(K\).
• In the exercise, we deal with two field extensions: \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\) and \(\mathbb{Q}[\sqrt{2} + \sqrt{3}]\).
• The basis of these extensions over \(\mathbb{Q}\) has four elements, and thus, their degree is 4.

Knowing the degree gives us insight into the "size" of the extension, namely how complex the field is compared to its base field.

Minimal Polynomial

The minimal polynomial is the polynomial of least degree that has a particular root within a field extension, and this polynomial plays a pivotal role in determining the degree of simple extensions.

• In the problem, we found that the extension \(\mathbb{Q}[\sqrt{2} + \sqrt{3}]\) is a simple extension where the element \(\sqrt{2} + \sqrt{3}\) is crucial.
• Its minimal polynomial is determined by considering powers of \(\sqrt{2} + \sqrt{3}\) and constructing a single polynomial that captures this extension. The minimal polynomial here is \(x^4 - 10x^2 + 1\).
• This polynomial is of degree 4, which is consistent with the calculated degree of the extension being 4, reaffirming the completeness of our chosen basis.

Choosing the correct minimal polynomial is essential as it influences the structure and properties of the extension field.

Vector Space Basis

A vector space basis in the context of field extensions refers to a set of elements in the extension that allows every other element in the space to be expressed as a linear combination of these basis elements with coefficients from the base field. Think of a basis like the framework that defines every possible combination in the field extension.

• For \(\mathbb{Q}[\sqrt{2}, \sqrt{3}]\), the basis we identified was \(\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}\). These elements are linearly independent and span the entire space.
• Similarly, when considering \(\mathbb{Q}[\sqrt{2} + \sqrt{3}]\), a similar set can describe it through algebraic manipulations.

These bases exemplify how field extensions can be systematically understood by translating algebraic roots into vector space concepts. The utility of having a basis is not just for finding degrees of extensions, but also for simplifying calculations involving elements of the extension.