N-Dimensional Real Division Algebras

[Math Amateur]

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...

I need help with understanding some remarks that Matej Bresar makes in Chapter 1 ...

The relevant text is as follows:View attachment 6216
View attachment 6217My questions regarding the above text are as follows:Question 1

In the above text from Bresar we read the following:

" ... ... Is it possible to define multiplication on an \(\displaystyle n\)-dimensional real space so that it becomes a real division algebra?

For \(\displaystyle n = 1\) the question is trivial; every element is a scalar multiple of unity and therefore up to an isomorphism \(\displaystyle \mathbb{R}\) itself is the only such algebra. ... ... "How do we know exactly (rigorously and formally) that up to an isomorphism \(\displaystyle \mathbb{R}\) itself is the only such algebra?

Question 2

In the above text from Bresar we read the following:

" ... ... for \(\displaystyle n = 2\) we know one example, \(\displaystyle \mathbb{C}\), but are there any other? This question is quite easy and the reader may try to solve it immediately. ... ... "

Can someone please help me to answer the above question posed by Bresar?

Question 3

In the above text from Bresar we read the following:

" ... ... what about \(\displaystyle n = 3\)? ... ... "Bresar answers this question on page 4 after proving Lemmas 1.1, 1.2, and 1.3 ...

Bresar writes:

" ... ... Lemma 1.3 rules out the case where \(\displaystyle n = 3\). ... ... "Can someone please help me to understand why/how Lemma 1.3 rules out the case where \(\displaystyle n = 3\)?
Lemma 1.3 and its proof read as follows:
View attachment 6218
Help with the above questions will be much appreciated ... ...

Peter
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So that readers of the above post can reference other parts of Bresar's arguments, Lemmas and proofs ... as well as appreciate the context of my questions I am providing pages 1-4 of Matej Bresar's book ... as follows:View attachment 6219
https://www.physicsforums.com/attachments/6220
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View attachment 6222

 

[jvicens]

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Page 1:

Introduction to Noncommutative Algebra

Chapter 1: Finite Dimensional Division Algebras

1.1. Introduction

Division algebras are a special class of algebras with an inverse element for every nonzero element. These algebras play an important role in mathematics and physics, and have been studied extensively. In this chapter, we will focus on finite dimensional division algebras over a field. This means that the algebra has a finite number of basis elements and is defined over a specific field.

1.2. Basic Definitions and Examples

Let F be a field, and let D be a division algebra over F. We say that D is finite dimensional if it has a finite basis over F. In this case, we say that D has dimension n over F and write dimF(D) = n. The elements of D are called scalars if they belong to F, and they are called vectors if they belong to D but not to F. We denote the set of scalars by F and the set of vectors by D*. Note that D* is a group under multiplication.

Example 1.1. The field of complex numbers \mathbb{C} is a division algebra over \mathbb{R} with dimension 2. The basis for \mathbb{C} over \mathbb{R} is {1, i}, where i^2 = -1.

Example 1.2. The field of quaternions \mathbb{H} is a division algebra over \mathbb{R} with dimension 4. The basis for \mathbb{H} over \mathbb{R} is {1, i, j, k}, where i^2 = j^2 = k^2 = -1 and ij = k, jk = i, and ki = j.

1.3. The Question of Existence

Given a field F, we can ask the following question: Is it possible to define multiplication on an n-dimensional real space so that it becomes a real division algebra? For n = 1 the question is trivial; every element is a scalar multiple of unity and therefore up to an isomorphism \mathbb{R} itself is the only such algebra. For n = 2 we know one example, \mathbb{C}, but are there any other? This question is quite easy and the