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I am reading W. Keith Nicholson's book: Introduction to Abstract Algebra (Third Edition) ...
I am focused on Section 4.2:Factorization of Polynomials over a Field.
I need some help with the proof of Part 1 of Theorem 12 on page 218
The relevant text from Nicholson's book is as follows:https://www.physicsforums.com/attachments/4596I do not follow how Nicholson is using the Principle of Mathematical Induction in Part (1) of the above proof ... ... indeed I find his proof strategy quite puzzling.
Nicholson seems to me to proceed as follows:
He proves the case for \(\displaystyle n\) = deg \(\displaystyle f = 1\) ... well and good ...
Now, according to my understanding of the Principle of Mathematical Induction he should assume (1) is true for deg \(\displaystyle f = n\) and then prove it true for deg \(\displaystyle f = n + 1\).
... ... BUT ... Nicholson does not seem to do this ... ?Can someone please explain exactly how Nicholson is using the Principle of Mathematical Induction ...
Help will be appreciated ...
Peter
I am focused on Section 4.2:Factorization of Polynomials over a Field.
I need some help with the proof of Part 1 of Theorem 12 on page 218
The relevant text from Nicholson's book is as follows:https://www.physicsforums.com/attachments/4596I do not follow how Nicholson is using the Principle of Mathematical Induction in Part (1) of the above proof ... ... indeed I find his proof strategy quite puzzling.
Nicholson seems to me to proceed as follows:
He proves the case for \(\displaystyle n\) = deg \(\displaystyle f = 1\) ... well and good ...
Now, according to my understanding of the Principle of Mathematical Induction he should assume (1) is true for deg \(\displaystyle f = n\) and then prove it true for deg \(\displaystyle f = n + 1\).
... ... BUT ... Nicholson does not seem to do this ... ?Can someone please explain exactly how Nicholson is using the Principle of Mathematical Induction ...
Help will be appreciated ...
Peter