Question

Sei \(K\) ein endlicher Körper. Betrachten Sie die Abbildung \(f: K^{*} \rightarrow K^{*}\), die definiert ist durch $$ f(x):=x^{2} $$ Ist \(f\) ein Gruppenhomomorphismus? (Wenn ja, zwischen welchen Gruppen?) Ist \(f\) ein Körperhomomorphismus? Geben Sie gegebenenfalls den Kern und das Bild von \(f\) an. [Unterscheiden Sie Körper, in denen \(1+1=0\) gilt, von allen anderen.]

Short answer

Yes, \( f \) is a group homomorphism between \( (K^*, \cdot) \). It is not a field homomorphism. The kernel is \( \{1, -1\} \) if \( 1 + 1 \neq 0 \), \( \{1\} \) if \( 1 + 1 = 0 \). The image depends on the field's characteristic.

Step-by-step solution

Understanding the Problem

We need to determine whether the function \( f: K^* \rightarrow K^* \), defined by \( f(x) = x^2 \), is a group homomorphism and/or a field homomorphism. Additionally, we need to find the kernel and image of \( f \) and consider special cases where \( 1 + 1 = 0 \).

Definition of Group Homomorphism

A function \( f: G \rightarrow H \) between groups \( G \) and \( H \) is a group homomorphism if for all \( a, b \in G \), \( f(ab) = f(a)f(b) \). We check if \( f(x) = x^2 \) satisfies this condition for \( K^* \), the multiplicative group of a field \( K \).

Checking Homomorphism Property

Let \( a, b \in K^* \). Then \( f(ab) = (ab)^2 = a^2b^2 = f(a)f(b) \), which shows \( f \) is a group homomorphism from \( K^* \) to \( K^* \).

Kernel of the Group Homomorphism

The kernel of a homomorphism \( f: G \rightarrow H \) is \( \{a \in G : f(a) = e_H\}\). Here, \( e_H = 1 \). We solve \( x^2 = 1 \) in \( K^* \), getting the solutions \( x = \pm 1 \) if \( 1 + 1 eq 0 \), and \( x = 1 \) if \( 1 + 1 = 0 \).

Image of the Group Homomorphism

The image of \( f \) is the set of all \( f(x) = x^2 \) for \( x \in K^* \), also known as the set of squares in \( K^* \). In characteristic \( 2 \), every element \( x \) satisfies \( x^2 = x \), so \( \text{Im}(f) = K^* \). In other cases, \( \text{Im}(f) \) generally includes half of the elements of \( K^* \).

Definition of Field Homomorphism

A function between fields is a homomorphism if it preserves both addition and multiplication. We test whether \( f(x + y) = f(x) + f(y) \), but since \( f(x) = x^2 \), this condition doesn't generally hold, especially as squaring doesn't preserve addition.

Conclusion of Homomorphism Types

\( f \) is a group homomorphism between the multiplicative groups \( (K^*, \cdot) \). It is not a field homomorphism because it does not preserve addition.

Result for Specific Fields

In fields where \( 1 + 1 = 0 \), the kernel is \( \{1\} \) and the image is \( K^* \). For other fields, the kernel is \( \{1, -1\} \), and the image consists of all squares in \( K^* \).

Key concepts

Finite Fields

Finite fields, often referred to as Galois fields, are mathematical structures with a finite number of elements. A field is a set equipped with two operations, usually referred to as addition and multiplication, satisfying certain properties such as associativity, commutativity, distributivity, and the existence of additive and multiplicative identities and inverses. In a finite field, all these operations are subject to a finite number of elements.

Finite fields are denoted as \( GF(p^n) \), where \( p \) is a prime number known as the characteristic of the field, and \( n \) is a positive integer indicating the degree of the field extension. The total number of elements in a finite field is given by \( p^n \). For instance, \( GF(2) \) consists of two elements \( \{0, 1\} \) and is often used to model binary operations in computer science and coding theory.

In the context of group homomorphisms, finite fields are particularly interesting because their multiplicative groups, denoted \( K^* \) (the set of non-zero elements under multiplication), are cyclic. This means that there exists an element, called a generator, from which every other element can be derived by iteratively multiplying the generator by itself. This property simplifies the verification of mappings like group homomorphisms, as any function between such cycles can be expressed in terms of its behavior on the generator.

Kernel and Image

The kernel and image of a homomorphism are critical concepts in group theory. The kernel of a group homomorphism \( f: G \rightarrow H \) is the set of elements in \( G \) that are mapped to the identity element of \( H \). Formally, it is denoted as \( \text{Ker}(f) = \{g \in G : f(g) = e_H\} \). In our specific scenario with the function \( f(x) = x^2 \), the kernel consists of those elements \( x \in K^* \) for which \( x^2 = 1 \).

Depending on the field, the kernel can vary. In fields where \( 1 + 1 = 0 \), known as fields of characteristic 2, \( -1 \equiv 1 \), therefore the kernel is \( \{1\} \). In other fields, it includes both \( \{1, -1\}\).

The image of a homomorphism, \( \text{Im}(f) \), is the set of all outputs resulting from elements in the domain being transformed by the function: \( \text{Im}(f) = \{f(g) : g \in G\} \). For the map \( f(x) = x^2 \) on a finite field, the image represents all the squares of the non-zero elements. In a field of characteristic 2, every element squared remains a part of \( K^* \), hence \( \text{Im}(f) = K^* \). In other cases, the image usually comprises about half the elements of \( K^* \), showing the diversity of possible squared outcomes.

Field Homomorphism

A field homomorphism is a function between two fields that respects both the structure of the addition and multiplication operations within those fields. To qualify as a field homomorphism, a mapping \( f: F \rightarrow L \) between fields must satisfy: \( f(x+y) = f(x) + f(y) \) and \( f(xy) = f(x)f(y) \) for all elements \( x, y \) in \( F \). Additionally, it must map the multiplicative identity of \( F \) to the multiplicative identity of \( L \).

In our exercise, the function \( f(x) = x^2 \) does not meet the criteria for being a field homomorphism. While it preserves multiplication, as seen in \( f(ab) = (ab)^2 = a^2b^2 = f(a)f(b) \), it fails to preserve addition. For instance, generally \( (a+b)^2 eq a^2 + b^2 \), which means \( f(a+b) eq f(a) + f(b) \). This violates a necessary condition for field homomorphisms.

Therefore, despite being a valid group homomorphism under multiplication, the function cannot qualify as a field homomorphism. This example highlights how specific operations like squaring can naturally align with certain structure-preserving criteria (like those of groups), yet diverge when applied to stricter conditions that account for additional operations, such as those defining field homomorphisms.