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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:
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
============================================================================
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 14 of Matej Bresar's book ... as follows:
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
============================================================================
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 14 of Matej Bresar's book ... as follows:
Attachments

Bresar  1  After the Complex Numbers  PART 1 ... ....png48.5 KB · Views: 695

Bresar  2  After the Complex Numbers  PART 2 ... ....png110.3 KB · Views: 736

Bresar  Lemma 1.3 ... ... .png71.8 KB · Views: 580

Bresar  Page 1.png34.7 KB · Views: 605

Bresar  Page 2.png66.8 KB · Views: 567

Bresar  Page 3 ... ....png41.3 KB · Views: 665

Bresar  Page 4 ... ....png38.8 KB · Views: 647