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I am reading Anderson and Feil - A First Course in Abstract Algebra.
I am currently focused on Ch. 45: The Splitting Field ... ...
I need some help with some aspects of the proof of Theorem 45.4 ...
Theorem 45.4 and its proof read as follows:View attachment 6677
My questions on the above proof are as follows:Question 1In the above text from Anderson and Feil we read the following:"... ... This means that $$f = ( x - \alpha)^k g$$, where $$k$$ is an integer greater than $$1$$ and $$g$$ is a polynomial over $$K$$ ... ... Since $$f$$ is in $$F[x]$$ ... that is $$f$$ is over $$F$$ ... shouldn't g be over $$F$$ not $$K$$?
(I am assuming that f being "over $$F$$" means the coefficients of $$f$$ are in $$F$$ ... )
Question 2In the above text from Anderson and Feil we read the following:"... ... We then have that $$x - \alpha$$ is a factor of both $$f$$ and $$f'$$. But if we use term-by-term differentiation instead, it is clear that $$f'\in F[x]$$. ... ... "What do Anderson and Feil mean by term-by-term differentiation in this context ... ... and if they do use term-by-term differentiation (what ever they mean) how does this show that $$f'\in F[x]$$ ... ... ?
Hope someone can help ...
Help will be much appreciated ... ...
Peter
I am currently focused on Ch. 45: The Splitting Field ... ...
I need some help with some aspects of the proof of Theorem 45.4 ...
Theorem 45.4 and its proof read as follows:View attachment 6677
My questions on the above proof are as follows:Question 1In the above text from Anderson and Feil we read the following:"... ... This means that $$f = ( x - \alpha)^k g$$, where $$k$$ is an integer greater than $$1$$ and $$g$$ is a polynomial over $$K$$ ... ... Since $$f$$ is in $$F[x]$$ ... that is $$f$$ is over $$F$$ ... shouldn't g be over $$F$$ not $$K$$?
(I am assuming that f being "over $$F$$" means the coefficients of $$f$$ are in $$F$$ ... )
Question 2In the above text from Anderson and Feil we read the following:"... ... We then have that $$x - \alpha$$ is a factor of both $$f$$ and $$f'$$. But if we use term-by-term differentiation instead, it is clear that $$f'\in F[x]$$. ... ... "What do Anderson and Feil mean by term-by-term differentiation in this context ... ... and if they do use term-by-term differentiation (what ever they mean) how does this show that $$f'\in F[x]$$ ... ... ?
Hope someone can help ...
Help will be much appreciated ... ...
Peter