Another Question Regarding Finite Dimensional Division Algebras - Bresar Lemma 1.1

[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 another aspect of the proof of Lemma 1.1 ... ...

Lemma 1.1 reads as follows:View attachment 6195
My questions regarding Bresar's proof above are as follows:Question 1

In the above text from Bresar, in the proof of Lemma 1.1 we read the following:

"... ... As we know, \(\displaystyle f( \omega )\) splits into linear and quadratic factors in \(\displaystyle \mathbb{R} [ \omega ]\) ... ..."

My question is ... how exactly do we know that \(\displaystyle f( \omega )\) splits into linear and quadratic factors in \(\displaystyle \mathbb{R} [ \omega ]\) ... can someone please explain this fact ... ...
Question 2

In the above text from Bresar, in the proof of Lemma 1.1 we read the following:

" ... ... Since \(\displaystyle f(x) = 0\) we have

\(\displaystyle ( x - \alpha_1 ) \ ... \ ... \ ( x - \alpha_r )( x^2 + \lambda_1 x + \mu_1 ) \ ... \ ... \ ( x^2 + \lambda_s x + \mu_s ) = 0 \)As \(\displaystyle D\) is a division algebra, one of the factors must be \(\displaystyle 0\). ... ... "My question is ... why does \(\displaystyle D\) being a division algebra mean that one of the factors must be zero ...?
Help with questions 1 and 2 above will be appreciated .. ...

Peter
=====================================================

In order for readers of the above post to appreciate the context of the post I am providing pages 1-2 of Bresar ... as follows ...View attachment 6196
https://www.physicsforums.com/attachments/6197

 

[jvicens]

f7b3e1d3a2c5e6d8-pdf.41328/Dear Peter,

Thank you for reaching out for help with your questions about Lemma 1.1 in Matej Bresar's book, "Introduction to Noncommutative Algebra." As a fellow scientist, I am happy to assist you in understanding this proof.

Question 1:

The statement "As we know, f(\omega) splits into linear and quadratic factors in \mathbb{R}[\omega]" is referring to the Fundamental Theorem of Algebra. This theorem states that any non-constant polynomial with complex coefficients can be factored into linear and quadratic factors. Since the coefficients of f(\omega) are in \mathbb{R}, the factorization will also be in \mathbb{R}[\omega]. Therefore, we know that f(\omega) can be written as a product of linear and quadratic factors in \mathbb{R}[\omega].

Question 2:

The statement "Since f(x) = 0 we have (x - \alpha_1) ... (x - \alpha_r)(x^2 + \lambda_1x + \mu_1) ... (x^2 + \lambda_sx + \mu_s) = 0" is using the fact that if a polynomial f(x) has roots \alpha_1, ..., \alpha_r, then f(x) can be factored into (x - \alpha_1) ... (x - \alpha_r). In this case, since f(x) = 0, we know that the polynomial on the left-hand side has roots \alpha_1, ..., \alpha_r.

Now, since D is a division algebra, we know that every non-zero element has an inverse. This means that if we multiply any non-zero element in D by another non-zero element, the result will always be non-zero. Therefore, if none of the factors (x - \alpha_i) or (x^2 + \lambda_jx + \mu_j) are equal to 0, then the product will also be non-zero. This contradicts the fact that we know the left-hand side is equal to 0. Therefore, at least one of the factors must be equal to 0.

I hope this helps clarify your questions about the proof of Lemma 1.1. If you have any further questions, please don't hesitate