I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...(adsbygoogle = window.adsbygoogle || []).push({});

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

The relevant text is as follows:

My 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 ##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. ... ... "

How do we know exactly (rigorously and formally) that up to an isomorphism ##\mathbb{R}## itself is the only such algebra?

Question 2

In the above text from Bresar we read the following:

" ... ... for ##n = 2## we know one example, ##\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 ##n = 3##? ... ... "

Bresar answers this question on page 4 after proving Lemmas 1.1, 1.2, and 1.3 ... (see uploads below)

Bresar writes:

" ... ... Lemma 1.3 rules out the case where ##n = 3##. ... ... "

Can someone please help me to understand why/how Lemma 1.3 rules out the case where ##n = 3##?

Lemma 1.3 and its proof read as follows:

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:

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# I N-Dimensional Real Division Algebras

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