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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:

My questions on the above proof are as follows:

In 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## ... )

In 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:

My questions on the above proof are as follows:

**Question 1**In 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 2**In 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