Problem 10 - A Course in Galois Theory

Question

Suppose that \(L: K\) is a Galois extension with Galois group \(G\). If \(x \in L\), let $$ \operatorname{tr}(x)=\sum_{\sigma \in G} \sigma(x) $$ Show that \(\mathrm{tr}\) is a \(K\)-linear mapping of \(L\) onto \(K\). The mapping tr is the trace. What is the effect of \(\mathrm{tr}\) on \(K\) if char \(K|| G \mid\) ?

Step-by-Step Solution

Answer

tr is K-linear and maps L onto K. If char K divides |G|, tr is zero on K.

Step 1: Understanding the Trace

The trace function \( \mathrm{tr}(x) = \sum_{\sigma \in G} \sigma(x) \) is a sum over all automorphisms in the Galois group \( G \). This means each \( \sigma(x) \) is an image of \( x \) under some automorphism \( \sigma \), and \( \mathrm{tr}(x) \) is the sum of these images.

Step 2: Proving K-linearity

To show \( \mathrm{tr} \) is \( K \)-linear, we need to demonstrate: \( \mathrm{tr}(a x + b y) = a \cdot \mathrm{tr}(x) + b \cdot \mathrm{tr}(y) \) for \( a, b \in K \) and \( x, y \in L \). Observe:\[ \mathrm{tr}(a x + b y) = \sum_{\sigma \in G} \sigma(a x + b y) = \sum_{\sigma \in G} (a \sigma(x) + b \sigma(y)) \]By linearity of \( \sigma \), this equals:\[ a \sum_{\sigma \in G} \sigma(x) + b \sum_{\sigma \in G} \sigma(y) = a \cdot \mathrm{tr}(x) + b \cdot \mathrm{tr}(y) \] Thus, \( \mathrm{tr} \) is \( K \)-linear.

Step 3: Showing Tr Maps Onto K

Since \( L: K \) is a Galois extension, every element in \( K \) is stable under the action of \( G \), and \( \mathrm{tr}(x) \) computes over all Galois conjugates, the resulting sum \( \mathrm{tr}(x) \) is fixed by \( G \), hence it lies in \( K \). Hence, \( \mathrm{tr} \) maps onto \( K \).

Step 4: Consideration of Trace Effect Considering Characteristic

If \( \operatorname{char}(K) = p \) divides \( |G| \), where \( p \) is a prime and \( |G| \) is the cardinality of the Galois group, then in fields of positive characteristic, the sum of an element repeated \( |G| \) times may be zero due to the fact \( \sum_{i=1}^{|G|} 1 = |G| \equiv 0 \mod p \). Therefore, in this case, \( \mathrm{tr}(x) = 0 \) for each \( x \in K \).

Key Concepts

Galois extension

A Galois extension is a special kind of field extension. It is named after the mathematician Évariste Galois. If you have two fields, say, \( L \) and \( K \), where \( L \) is an extension of \( K \), we call it a Galois extension if

• it is a normal extension, and
• it is a separable extension.

Being a normal extension means every polynomial that has a root in \( L \) completely factors in \( L \). Separable means each polynomial has distinct roots. For a field to be a Galois extension, both conditions must be satisfied over \( K \). This ensures all symmetry of polynomial solutions is captured in the extension. Understanding these conditions helps to explore the symmetry aspects in field theory.

Galois group

The Galois group takes center stage when discussing Galois extensions. It is a group consisting of all automorphisms (bijective maps from a field onto itself that preserve operations such as addition and multiplication) of \( L \) that fix every element of \( K \). In symbols, each automorphism \( \sigma \) in the Galois group \( G \) satisfies \( \sigma(k) = k \) for all \( k \in K \).
This group captures the symmetries of the field extension; think of it like shuffling the roots of a polynomial and seeing which arrangements preserve the structure. It is a crucial tool for understanding how fields can be constructed and related.

Trace map

The trace map, sometimes simply called 'trace', is an important function in field theory. In the context of a Galois extension \( L: K \) with Galois group \( G \), the trace of an element \( x \in L \) is given by \[\mathrm{tr}(x) = \sum_{\sigma \in G} \sigma(x)\]This sum takes all the images of \( x \) under each automorphism in the Galois group and adds them up. The result is an element in \( K \), showing how \( x \) and properties of \( L \) get reflected back in \( K \). The trace not only indicates how elements of \( L \) are connected to \( K \), due to the symmetry captured by \( G \), but also helps in calculations within field theory contexts, such as in the study of bilinear forms and determinants.

K-linearity

K-linearity refers to the characteristic property of certain maps, like the trace map, that follow the rules of linearity using elements from the field \( K \). For a mapping to be \( K \)-linear, like the trace \( \mathrm{tr} \), it must satisfy the linearity condition:\[\mathrm{tr}(a x + b y) = a \cdot \mathrm{tr}(x) + b \cdot \mathrm{tr}(y) \text{ for } a, b \in K \text{ and } x, y \in L\]What this means is, if you scale or sum two elements in the extended field \( L \), the trace operation responds in a predictable way, proportionally scaling or summing the traces. This linear property makes trace mappings powerful for simplifying calculations, particularly when working with larger and more complex field extensions.

Field characteristic

The characteristic of a field, denoted \( \operatorname{char}(K) \), is a fundamental property that can be either 0 or a prime number \( p \). It dictates the behavior of addition in the field. If the characteristic is 0, it implies there's no smallest positive integer \( n \) where \( n \cdot 1 = 0 \).
If \( \operatorname{char}(K) = p \), the field "wraps around" upon adding 1, \( p \) times, making \( p \cdot 1 = 0 \). This affects calculations significantly, such as with the trace when \( p \) divides the order of \( G \), leading to the trace of elements possibly being zero, offering unique insights into the field's structure in different scenarios. Understanding a field's characteristic is crucial in determining how its arithmetic properties unfold.