Problem 13
Question
Let \(\widetilde{\mathbb{C}}\) be another field of complex numbers. Determine all mappings \(\varphi: \mathbb{C} \rightarrow \widetilde{\mathbb{C}}\) with the following properties; \(\begin{aligned}(a) & \varphi(z+w) &=\varphi(z)+\varphi(w) & & \text { for all } z, w \in \mathbb{C} \\\\(b) & \varphi(z w) &=\varphi(z) \varphi(w) & & \text { for all } z, w \in \mathbb{C} \\\\(c) & \varphi(x) &=x & & \text { for all } x \in \mathbb{R} \end{aligned}\) Remark. It turns out that such mappings exist, and they are automatically bijective; thus they give isomorphisms \(\mathbb{C} \rightarrow \widetilde{\mathbb{C}}\) that leave \(\mathbb{R}\) fixed element by element. The field of complex numbers is therefore essentially uniquely determined. In the special case \(\mathbb{C}=\widetilde{\mathbb{C}}\) we get automorphisms of \(\mathbb{C}\) with the fixed field \(\mathbb{R}\). Remark: What automorphisms (i.e. isomorphisms with itself) does the real field \(\mathbb{R}\) have ? Hint: Such an automorphism of \(\mathbb{R}\) must preserve the ordering of \(\mathbb{R} !\)
Step-by-Step Solution
Verified Answer
The mappings are identity and complex conjugation for complex numbers, with the identity as the sole automorphism for real numbers.
1Step 1: Understand the Properties of the Mapping
We need to find mappings \( \varphi: \mathbb{C} \rightarrow \widetilde{\mathbb{C}} \) such that they are linear (property (a)) and multiplicative (property (b)). Additionally, they must act as the identity on the real numbers (property (c)). These mappings are isomorphisms, which imply that they preserve both structure and order.
2Step 2: Consider Property (c) - Fixes Real Numbers
For any real number \( x \), the mapping \( \varphi(x) = x \). This means for any real number the mapping does not change the value, adhering strictly to the identity mapping on real numbers.
3Step 3: Analyze Linear Property (a) - Additive Homomorphism
The mapping satisfies \( \varphi(z+w) = \varphi(z) + \varphi(w) \) for all complex numbers \( z \) and \( w \). This property ensures linearity in terms of addition, much like a linear function.
4Step 4: Analyze Multiplicative Property (b) - Multiplicative Homomorphism
The mapping satisfies \( \varphi(zw) = \varphi(z) \varphi(w) \) for all complex numbers \( z \) and \( w \). This ensures compatibility with multiplication, similar to a homomorphism.
5Step 5: Explore Automorphisms of \(\mathbb{C}\) with \(\mathbb{R}\) Fixed
We are considering automorphisms of \( \mathbb{C} \) that leave each real number unchanged. In particular, this involves determining all possible transformations that satisfy all three properties.
6Step 6: Use Field Properties to Find Solutions
The known mappings that satisfy these properties in \( \mathbb{C} \) are the identity mapping and complex conjugation. Complex conjugation is defined for any complex number \( z = a + bi \) as \( \overline{z} = a - bi \), and it respects all three properties.
7Step 7: Determine Automorphisms of \(\mathbb{R}\)
For real numbers, the only mapping that preserves ordering and satisfies all automorphism properties is the identity mapping due to the order-preserving requirement.
Key Concepts
Complex AutomorphismsAdditive HomomorphismMultiplicative HomomorphismReal Field Automorphisms
Complex Automorphisms
In the realm of mathematics, particularly in complex analysis and field theory, a complex automorphism is an isomorphism of a complex field with itself. This involves transforming complex numbers in a way that the structure and operations of complex numbers are preserved.
For example, let's consider the complex number field \(\mathbb{C}\). The automorphisms of \(\mathbb{C}\) are special because they must satisfy three specific properties: they must be additive homomorphisms, multiplicative homomorphisms, and fix the field of real numbers \(\mathbb{R}\) element by element.
The two primary examples of complex automorphisms are:
For example, let's consider the complex number field \(\mathbb{C}\). The automorphisms of \(\mathbb{C}\) are special because they must satisfy three specific properties: they must be additive homomorphisms, multiplicative homomorphisms, and fix the field of real numbers \(\mathbb{R}\) element by element.
The two primary examples of complex automorphisms are:
- **Identity Mapping:** This simply maps every complex number to itself, \(\varphi(z) = z\).
- **Complex Conjugation:** This involves flipping the sign of the imaginary part of a complex number, so \(z = a + bi\) becomes \(\overline{z} = a - bi\).
Additive Homomorphism
An additive homomorphism is a type of mapping that respects addition operations between numbers. In other words, it's a way of ensuring that the sum of two elements is maintained after the transformation.
For our mapping \( \varphi : \mathbb{C} \rightarrow \widetilde{\mathbb{C}} \), the additive property means that \( \varphi(z + w) = \varphi(z) + \varphi(w) \) for any complex numbers \(z\) and \(w\).
This means that the map preserves linearity - it's additive just like a classic linear function that you might be familiar with from real analysis.
To understand this better, imagine you are working with vectors in a space. If you add two vectors together, an additive homomorphism ensures that the resulting vector after transformation is the same as individually transformed vectors added together.
This property is crucial because it ensures the map acts consistently on combined numbers, preserving the essential arithmetic structure of the complex field.
For our mapping \( \varphi : \mathbb{C} \rightarrow \widetilde{\mathbb{C}} \), the additive property means that \( \varphi(z + w) = \varphi(z) + \varphi(w) \) for any complex numbers \(z\) and \(w\).
This means that the map preserves linearity - it's additive just like a classic linear function that you might be familiar with from real analysis.
To understand this better, imagine you are working with vectors in a space. If you add two vectors together, an additive homomorphism ensures that the resulting vector after transformation is the same as individually transformed vectors added together.
This property is crucial because it ensures the map acts consistently on combined numbers, preserving the essential arithmetic structure of the complex field.
Multiplicative Homomorphism
A multiplicative homomorphism is a mapping that respects the multiplication operation. For the mapping \( \varphi: \mathbb{C} \rightarrow \widetilde{\mathbb{C}} \), the multiplicativity requirement is expressed as \( \varphi(zw) = \varphi(z) \varphi(w) \), where \(z\) and \(w\) are complex numbers.
This relationship is vital because it ensures the preservation of the product of numbers under the mapping.
Think about this property like multiplying fractions: if you have a mapping that doubles every fraction, the product of two fractions after transformation will equal the product before transformation, just doubled.
Together with additive homomorphism, multiplicative homomorphism assures that essential binary operations in mathematics - addition and multiplication - maintain consistency through transformations. This uniformity allows us to consider these mappings as isomorphisms, since the fundamental operations of the field remain intact.
This relationship is vital because it ensures the preservation of the product of numbers under the mapping.
Think about this property like multiplying fractions: if you have a mapping that doubles every fraction, the product of two fractions after transformation will equal the product before transformation, just doubled.
Together with additive homomorphism, multiplicative homomorphism assures that essential binary operations in mathematics - addition and multiplication - maintain consistency through transformations. This uniformity allows us to consider these mappings as isomorphisms, since the fundamental operations of the field remain intact.
Real Field Automorphisms
When examining real field automorphisms, we are delving into how transformations operate within the field of real numbers \(\mathbb{R}\). An automorphism here must satisfy several strict criteria to be valid.
Crucially, it must preserve order because real numbers have a natural order with respect to size. This requirement means that just about the only automorphism available in the field of real numbers is the identity mapping, where each number maps to itself.
Why is this? Consider if the order was not preserved—this would result in inconsistencies and contradictions in mathematical operations and relationships. For instance, if 3 was not greater than 2, our understanding of greater and lesser would not hold, leading to a breakdown of the entire real number system.
Thus, within the realm of real field automorphisms, while it might seem limiting, the preservation of order ensures the reliability and logical consistency of all mathematical operations performed with real numbers.
Crucially, it must preserve order because real numbers have a natural order with respect to size. This requirement means that just about the only automorphism available in the field of real numbers is the identity mapping, where each number maps to itself.
Why is this? Consider if the order was not preserved—this would result in inconsistencies and contradictions in mathematical operations and relationships. For instance, if 3 was not greater than 2, our understanding of greater and lesser would not hold, leading to a breakdown of the entire real number system.
Thus, within the realm of real field automorphisms, while it might seem limiting, the preservation of order ensures the reliability and logical consistency of all mathematical operations performed with real numbers.
Other exercises in this chapter
Problem 12
For \(k \in \mathbb{N}_{0}\), and \(z \in \mathbb{C}\) with \(|z|
View solution Problem 12
There is no continuous function \(l: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}\) such that $$ \exp (l(z))=z \text { for all } z \in \mathbb{C}^{\bullet} $$
View solution Problem 13
Let \(\left(a_{n}\right)_{n \geq 0}\) and \(\left(b_{n}\right)_{n \geq 0}\) be two sequences of complex numbers and $$ A_{n}:=a_{0}+a_{1}+\cdots+a_{n}, \quad n
View solution Problem 13
Let \(n \geq 2\) be a natural number. There is no continuous function \(f: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}^{\bullet}\) with the two properties (a) \
View solution