Understanding Non-Galois Field Extensions: The Case of (mathbb{Q}sqrt[3]{2}) and (mathbb{Q}sqrt[4]{2})

In the field of abstract algebra, a Galois extension is a fundamental concept. However, not all field extensions have this property. In this article, we will explore the non-Galois extensions, leveraging two well-known examples: (mathbb{Q}sqrt[3]{2}) over (mathbb{Q}) and (mathbb{Q}sqrt[4]{2}) over (mathbb{Q}).

Introduction to Field Extensions

A field extension (mathbb{L}/mathbb{K}) is defined when (mathbb{K}) is a subfield of the field (mathbb{L}). The inclusion map (mathbb{K} hookrightarrow mathbb{L}) allows elements of (mathbb{K}) to be thought of as elements of (mathbb{L}). For instance, (mathbb{Q}sqrt[3]{2}) is the field of all numbers of the form (a bsqrt[3]{2} csqrt[3]{4}) where (a, b, c in mathbb{Q}).

Example 1: (mathbb{Q}sqrt[3]{2}) Over (mathbb{Q})

The field (mathbb{Q}sqrt[3]{2}) is an extension of (mathbb{Q}) provided by the inclusion of (sqrt[3]{2}). The minimal polynomial of (sqrt[3]{2}) over (mathbb{Q}) is (x^3 - 2), which is irreducible over (mathbb{Q}). Hence, the degree of this extension is ([ mathbb{Q}sqrt[3]{2} : mathbb{Q} ] 3).

Despite this, (mathbb{Q}sqrt[3]{2}) is not a Galois extension of (mathbb{Q}). For an extension to be Galois, it must be both normal and separable. Being normal means that (mathbb{Q}sqrt[3]{2}) must contain all the roots of the polynomial (x^3 - 2). However, the roots of (x^3 - 2) are (sqrt[3]{2}), (sqrt[3]{2}omega), and (sqrt[3]{2}omega^2), where (omega) is a primitive cube root of unity. Notice that (sqrt[3]{2}omega) and (sqrt[3]{2}omega^2) are not elements of (mathbb{Q}sqrt[3]{2}).

Recall that (omega e^{2pi i/3}), and therefore, (sqrt[3]{2}omega) and (sqrt[3]{2}omega^2) are complex numbers. Since (mathbb{Q}sqrt[3]{2}) is a real field and does not contain these complex roots, it fails to be normal. Hence, (mathbb{Q}sqrt[3]{2}) is not a Galois extension of (mathbb{Q}).

Example 2: (mathbb{Q}sqrt[4]{2}) Over (mathbb{Q})

The polynomial (x^4 - 2) has roots in the complex numbers, specifically (sqrt[4]{2}, -sqrt[4]{2}, sqrt[4]{2}i), and (-sqrt[4]{2}i). Consider the polynomial (x^4 - 2) over (mathbb{Q}). If (mathbb{Q}sqrt[4]{2}) were a Galois extension, (x^4 - 2) would split completely into linear factors over (mathbb{Q}sqrt[4]{2}).

However, it is clear that (mathbb{Q}sqrt[4]{2}) is a subfield of (mathbb{R}). Therefore, (-sqrt[4]{2}i) and (sqrt[4]{2}i), the non-real roots of (x^4 - 2), cannot be in (mathbb{Q}sqrt[4]{2}). This shows that (x^4 - 2) does not split into linear factors over (mathbb{Q}sqrt[4]{2}), illustrating that (mathbb{Q}sqrt[4]{2}) is not a normal extension, and hence not a Galois extension of (mathbb{Q}).

Conclusion

Through these examples, we have seen how the concepts of normal and separable extensions come into play in the realm of Galois theory. Understanding non-Galois extensions is crucial for a deeper comprehension of field theory and algebraic structures. By exploring the specific properties of these extensions, we gain insight into the intricacies of field extensions and their classification as either Galois or non-Galois.