Problem 2 - Abstract Algebra: Theory and Applications

Question

Find a basis for each of the following field extensions. What is the degree of each extension? (a) \(Q(\sqrt{3}, \sqrt{6})\) over \(\mathbb{Q}\) (b) \(\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{3})\) over \(\mathbf{Q}\) (c) \(Q(\sqrt{2}, i)\) over \(Q\) (d) \(Q(\sqrt{3}, \sqrt{5}, \sqrt{7})\) over \(\mathbb{Q}\) (e) \(\mathrm{Q}(\sqrt{2}, \sqrt[3]{2})\) over \(\mathrm{Q}\) (f) \(Q(\sqrt{8})\) over \(Q(\sqrt{2})\) (g) \(\mathbb{Q}(i, \sqrt{2}+i, \sqrt{3}+i)\) over \(\mathbb{Q}\) (h) \(Q(\sqrt{2}+\sqrt{5})\) over \(Q(\sqrt{5})\) (i) \(\mathbb{Q}(\sqrt{2}, \sqrt{6}+\sqrt{10})\) over \(Q(\sqrt{3}+\sqrt{5})\)

Step-by-Step Solution

Answer

The basis for \(Q(\sqrt{3},\sqrt{6})\) over \(\mathbb{Q}\) is \(\{1, \sqrt{3}, \sqrt{2}, \sqrt{6}\}\). 2. What is the degree of \(Q(\sqrt{2}, i)\) over \(Q\)? The degree of \(Q(\sqrt{2}, i)\) over \(Q\) is \(4\). 3. What is the basis for \(Q(\sqrt{2}+\sqrt{5})\) over \(Q(\sqrt{5})\)? The basis for \(Q(\sqrt{2}+\sqrt{5})\) over \(Q(\sqrt{5})\) is \(\{1, \sqrt{5}, \sqrt{2}+\sqrt{5}\}\). 4. What is the degree of \(Q(\sqrt{2}, \sqrt{6}+\sqrt{10})\) over \(Q(\sqrt{3}+\sqrt{5})\)? The degree of \(Q(\sqrt{2}, \sqrt{6}+\sqrt{10})\) over \(Q(\sqrt{3}+\sqrt{5})\) is \(3\).

Step 1:

: We will find a basis for this extension by finding the simplest form of elements that can form this extension. We have \(\sqrt{3}\) and \(\sqrt{6}\), which can be broken down into the elements for \(\sqrt{3}\) and \(\sqrt{2}\). In the extension, we can have elements in the form \(a+b\sqrt{3}+c\sqrt{2}+d\sqrt{6}\), where \(a,b,c,d\in\mathbb{Q}\). The basis for this extension is found to be \(\{1, \sqrt{3}, \sqrt{2}, \sqrt{6}\}\). Determining the degree of the extension

Step 2:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(4\). (b) Finding a basis for \(\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{3})\) over \(\mathbf{Q}\)

Step 3:

: The elements of the form \(a+b\sqrt[3]{2}+c\sqrt[3]{4}+d\sqrt[3]{3}+e\sqrt[3]{9}+f\sqrt[3]{6}\) can be found in the extension, where \(a,b,c,d,e,f\in\mathbb{Q}\). The basis for this extension is found to be \(\{1, \sqrt[3]{2}, \sqrt[3]{4}, \sqrt[3]{3}, \sqrt[3]{9}, \sqrt[3]{6}\}\). Determining the degree of the extension

Step 4:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(6\). (c) Finding a basis for \(Q(\sqrt{2}, i)\) over \(Q\)

Step 5:

: We have elements of the form \(a+b\sqrt{2}+ci+di\sqrt{2}\) in the extension, where \(a,b,c,d\in\mathbb{Q}\). The basis for this extension is found to be \(\{1, \sqrt{2}, i, i\sqrt{2}\}\). Determining the degree of the extension

Step 6:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(4\). (d) Finding a basis for \(Q(\sqrt{3}, \sqrt{5}, \sqrt{7})\) over \(\mathbb{Q}\)

Step 7:

: In this extension, we can have elements of the form \(a+b\sqrt{3}+c\sqrt{5}+d\sqrt{7}+e\sqrt{15}+f\sqrt{21}+g\sqrt{35}+h\sqrt{105}\), where \(a,b,c,d,e,f,g,h\in\mathbb{Q}\). The basis for this extension is \(\{1, \sqrt{3}, \sqrt{5}, \sqrt{7}, \sqrt{15}, \sqrt{21}, \sqrt{35}, \sqrt{105}\}\). Determining the degree of the extension

Step 8:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(8\). (e) Finding a basis for \(\mathrm{Q}(\sqrt{2}, \sqrt[3]{2})\) over \(\mathrm{Q}\)

Step 9:

: The elements of the form \(a+b\sqrt{2}+c\sqrt[3]{2}+d\sqrt[3]{4}+e\sqrt{2}\sqrt[3]{2}+f\sqrt{2}\sqrt[3]{4}\) can be found in the extension, where \(a,b,c,d,e,f\in\mathbb{Q}\). The basis for this extension is found to be \(\{1, \sqrt{2}, \sqrt[3]{2}, \sqrt[3]{4}, \sqrt{2}\sqrt[3]{2}, \sqrt{2}\sqrt[3]{4}\}\). Determining the degree of the extension

Step 10:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(6\). (f) Finding a basis for \(Q(\sqrt{8})\) over \(Q(\sqrt{2})\)

Step 11:

: We can rewrite \(\sqrt{8}\) as \(2\sqrt{2}\). Thus, \(Q(\sqrt{8}) \equiv Q(2\sqrt{2})\). Since \(2\sqrt{2} = 2(\sqrt{2})\), the basis for this extension is simply \(\{1\}\). Determining the degree of the extension

Step 12:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(1\). (g) Finding a basis for \(\mathbb{Q}(i, \sqrt{2}+i, \sqrt{3}+i)\) over \(\mathbb{Q}\)

Step 13:

: In this extension, we can find elements in the form \(a+bi+ci\sqrt{2}+di\sqrt{3}\) where \(a,b,c,d\in\mathbb{Q}\). The basis for this extension is found to be \(\{1, i, i\sqrt{2}, i\sqrt{3}\}\). Determining the degree of the extension

Step 14:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(4\). (h) Finding a basis for \(Q(\sqrt{2}+\sqrt{5})\) over \(Q(\sqrt{5})\)

Step 15:

: We can have elements of the form \(a+b\sqrt{5}+c(\sqrt{2}+\sqrt{5})\) where \(a, b, c\in\mathbb{Q}\). The basis for this extension is found to be \(\{1, \sqrt{5}, \sqrt{2}+\sqrt{5}\}\). Determining the degree of the extension

Step 16:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(3\). (i) Finding a basis for \(\mathbb{Q}(\sqrt{2}, \sqrt{6}+\sqrt{10})\) over \(Q(\sqrt{3}+\sqrt{5})\)

Step 17:

: Note that \(\sqrt{6}+\sqrt{10} = \sqrt{2}\sqrt{3}+\sqrt{2}\sqrt{5}\), which is a product of existing elements. Therefore, the basis for this extension is the set \(\{1, \sqrt{2}, \sqrt{3}+\sqrt{5}\}\). Determining the degree of the extension

Step 18:

: The degree of the extension is equal to the number of elements in the basis. In this case, the degree is \(3\).

Key Concepts

Basis of Field Extension

Understanding the basis of a field extension is crucial in algebra, as it provides the framework to add structure to an otherwise abstract concept. In essence, a basis is a set of elements in an extended field that, when combined with the field's operations, can express any element within the extension. Think of it like constructing a variety of buildings (elements of the extended field) using a limited set of blueprints (basis elements).

For example, in the extension \( \mathbb{Q}(\sqrt{3}, \sqrt{6}) \) over \( \mathbb{Q} \), the basis is \( \{1, \sqrt{3}, \sqrt{2}, \sqrt{6}\} \). These 'blueprints' enable us to construct any element within the field extension by creating linear combinations with coefficients from the rational numbers. Each element of the basis is like a unique building block that, when put together in various combinations, creates the entire cityscape of the field extension.

Degree of Field Extension

To appreciate the complexity of a field extension, think of the degree as measuring the 'size' of the extension in comparison to the original field. Specifically, the degree is the number of basis elements. It tells us how much bigger the extended field is relative to the original field, much like comparing the size of two nesting dolls.

Using our earlier example of \( \mathbb{Q}(\sqrt{3}, \sqrt{6}) \) over \( \mathbb{Q} \), since the basis contains four elements, the degree of the field extension is \(4\). This means the extended field is, in a sense, four times 'larger' or more complex than the original field of rational numbers. Higher-degree extensions are akin to more intricate and expansive structures, offering a greater variety of combinations of the basis elements.

Rational Numbers \( Q \)

Rational numbers, symbolized as \( \mathbb{Q} \), are the set of numbers that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b \eq 0\). They form the foundation of more advanced number systems, comparable to the basic ingredients in a recipe. Through field extensions, we can 'cook up' more complex number systems by adding new elements like square roots or complex numbers.

When we extend the rational numbers by adding new elements, such as \( \sqrt{2} \) or \( i \), we step into a larger world of numbers that cannot be neatly expressed as simple fractions. These field extensions are like expanding our recipe to create more elaborate dishes out of the basic ingredients.

Root Elements

Root elements are essentially the 'x-factor' that, when added to a field (like the rational numbers), create a field extension. They are the new elements, such as square roots or cube roots, that were not previously in the field.

For instance, when we take \( \mathbb{Q} \) and introduce \( \sqrt{2} \) into the mix, we dive into the realm of irrational numbers, which cannot be represented as fractions of integers. These root elements enable us to explore equations that would otherwise be unsolvable within the confines of \( \mathbb{Q} \). Adding \( \sqrt{2} \) is like discovering a new spice that transforms the taste profile of our basic recipe into something entirely new and exciting.