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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 the proof of Lemma 1.25 ...
Lemma 1.25 reads as follows:
My questions on the proof of Lemma 1.25 are as follows:Question 1
In the above text from Bresar we read the following:
" ... ... Therefore ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]## ... ... "Can someone please explain exactly why Bresar is concluding that ##[ M(A) \ : \ F ] \ge d^2## ... ... ?Question 2
In the above text from Bresar we read the following:
" ... ... Therefore ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]##
and so ##M(A) = [ \text{ End}_F (A) \ : \ F ]##. ... ... "Can someone please explain exactly why ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]## ... ...
... implies that ... ##M(A) = [ \text{ End}_F (A) \ : \ F ]## ...
Hope someone can help ...
Peter
===========================================================*** NOTE ***
So that readers of the above post will be able to understand the context and notation of the post ... I am providing Bresar's first two pages on Multiplication Algebras ... ... as follows:
I need help with the proof of Lemma 1.25 ...
Lemma 1.25 reads as follows:
My questions on the proof of Lemma 1.25 are as follows:Question 1
In the above text from Bresar we read the following:
" ... ... Therefore ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]## ... ... "Can someone please explain exactly why Bresar is concluding that ##[ M(A) \ : \ F ] \ge d^2## ... ... ?Question 2
In the above text from Bresar we read the following:
" ... ... Therefore ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]##
and so ##M(A) = [ \text{ End}_F (A) \ : \ F ]##. ... ... "Can someone please explain exactly why ##[ M(A) \ : \ F ] \ge d^2 = [ \text{ End}_F (A) \ : \ F ]## ... ...
... implies that ... ##M(A) = [ \text{ End}_F (A) \ : \ F ]## ...
Hope someone can help ...
Peter
===========================================================*** NOTE ***
So that readers of the above post will be able to understand the context and notation of the post ... I am providing Bresar's first two pages on Multiplication Algebras ... ... as follows:
Attachments

Bresar  1  Lemma 1.25  PART 1 ... ....png27.6 KB · Views: 574

Bresar  2  Lemma 1.25  PART 2 ... ....png23.9 KB · Views: 687

Bresar  1  Section 1.5 Multiplication Algebra  PART 1 ... ....png27.6 KB · Views: 532

Bresar  2  Section 1.5 Multiplication Algebra  PART 2 ... ....png29.5 KB · Views: 714
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