Problem 13 - A Course in Galois Theory

Question

(Hilbert's theorem 90) Suppose that \(L: K\) is a Galois extension of degree \(n\) with cyclic Galois group generated by \(\tau\), say. (i) Suppose that \(\alpha=\beta / \tau(\beta)\). Show that \(N(\alpha)=1\). (ii) Suppose that \(N(\alpha)=1 .\) Let \(c_{0}=\alpha, c_{1}=\alpha \tau\left(c_{0}\right), c_{2}=\alpha \tau\left(c_{1}\right), \ldots\), \(c_{n-1}=\alpha \tau\left(c_{n-2}\right) .\) Show that there exists \(\gamma\) in \(L\) such that \(\beta=c_{0} \gamma+c_{1} \tau(\gamma)+\cdots+c_{n-1} \tau^{n-1}(\gamma) \neq 0\) Show that \(\alpha=\beta / \tau(\beta)\).

Step-by-Step Solution

Answer

(i) \(N(\alpha)=1\). (ii) Exists \(\gamma\) such that \(\beta = \sum c_i \tau^i(\gamma)\).

Step 1: Understanding the Norm in Galois Extensions

Given a Galois extension \(L:K\) with a cyclic Galois group generated by \(\tau\), the norm \(N(\alpha)\) is defined as \(\alpha \cdot \tau(\alpha) \cdot \tau^2(\alpha) \cdot \ldots \cdot \tau^{n-1}(\alpha)\). Our goal is to show that if \(\alpha = \beta / \tau(\beta)\), then \(N(\alpha) = 1\).

Step 2: Show \(N(\alpha) = 1\) for Part (i)

Since \(\alpha = \beta / \tau(\beta)\), let us calculate the norm: \[N(\alpha) = \alpha \cdot \tau(\alpha) \cdot \tau^2(\alpha) \cdot \ldots \cdot \tau^{n-1}(\alpha)\]. Replacing \(\alpha\) by \(\beta / \tau(\beta)\), this becomes \[N(\alpha) = \frac{\beta}{\tau(\beta)} \cdot \frac{\tau(\beta)}{\tau^2(\beta)} \cdot \ldots \cdot \frac{\tau^{n-1}(\beta)}{\beta}\]. The terms cancel out, leaving \(N(\alpha) = 1\).

Step 3: Define Sequence for Part (ii)

Create the sequence \(c_0 = \alpha, c_1 = \alpha\tau(c_0), c_2 = \alpha\tau(c_1), \ldots, c_{n-1} = \alpha\tau(c_{n-2})\). By exploring this recursively, notice that \(c_i = \alpha \tau(c_{i-1}) = \beta / \tau^{i+1}(\beta)\).

Step 4: Existence of \(\gamma\) in Part (ii)

To show the existence of \(\gamma \in L\) such that \(\beta = c_0 \gamma + c_1 \tau(\gamma) + \cdots + c_{n-1} \tau^{n-1}(\gamma) eq 0\), observe that it requires finding a \(\gamma\) that satisfies the equation given earlier. Since the determinant of the associated matrix transformation derived from \(c_i\)'s is related to the norm \(N(\alpha) = 1\), and hence invertible, there exists such \(\gamma\).

Step 5: Concluding Part (ii) Solution

Since there exists such a \(\gamma\), substituting back gives the form \(\beta = \sum_{i=0}^{n-1}c_i\tau^i(\gamma)\), effectively showing the requested relation in \(L\). Thus, the construction of \(\beta\) guarantees \(\alpha = \beta / \tau(\beta)\) holds, completing the second part of the problem.

Key Concepts

Cyclic Galois Group

In mathematics, especially in the field of Galois theory, a cyclic Galois group is a group that can be generated by a single element, meaning every element of the group can be composed as some power of this single generator. This concept is crucial when working with field extensions because it simplifies the analysis of field symmetries.

A Galois extension, when its Galois group is cyclic, can be related to simple cyclic processes. For instance, if you have a Galois extension \(L:K\) with a cyclic Galois group of degree \(n\), it means that there are exactly \(n\) field automorphisms, and every one of them can be expressed as a power of a fundamental automorphism, often denoted \(\tau\).

This simplification allows for straightforward calculations such as the Galois norm and proves the basis for deeper explorations like those seen in Hilbert's Theorem 90.

Norm in Galois Extensions

The norm in Galois extensions is a way to "compress" information about an element \(\alpha\) in the extension field \(L\) into the base field \(K\). It is computed as a product of images of \(\alpha\) under all automorphisms in the Galois group. If the Galois group is cyclic, this is simply the product \(\alpha \cdot \tau(\alpha) \cdot \tau^2(\alpha) \cdot \ldots \cdot \tau^{n-1}(\alpha)\).

The role of the norm is significant in determining properties of field extensions. For example, if \( N(\alpha) = 1 \), particularly in cyclic extensions, it can be deduced through identities that depend on the unique nature of cyclic operations—like seeing how terms cancel out to result in 1 in calculations.

This concept is not only pivotal for solving problems like those proposed by Hilbert but it also provides a universally applicable method of transitioning between different compositional frameworks within Galois theory.

Hilbert's Theorem 90

Hilbert's Theorem 90 offers a particularly elegant simplification in the study of cyclic Galois extensions. The theorem states that for a Galois extension \(L:K\) with a cyclic Galois group, if the norm \(N(\alpha) = 1\), then there exists some element \(\beta\) such that \(\alpha = \beta / \tau(\beta)\).

This theorem has far-reaching implications in fields such as algebraic number theory and class field theory. It is applied to understand the structure of field extensions and provides methods to construct certain elements within a field. The significance of finding such a \(\beta\) lies in demonstrating that describing an element within a Galois extension can often reduce to a straightforward division of two terms,

following sequences generated by group automorphisms. Part (ii) of the problem you are tackling is a direct application of Hilbert's Theorem 90, ensuring that order and norm functions intertwine seamlessly, guaranteeing the structural consistency of cyclic Galois groups.