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Mathematics
Linear and Abstract Algebra
Splitting Fields: Anderson and Feil, Theorem 45.4 ....
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[QUOTE="Math Amateur, post: 5787778, member: 203675"] 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: [ATTACH=full]205841[/ATTACH] My questions on the above proof are as follows:[I][B]Question 1[/B][/I]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## ... ) [U][I][B]Question 2[/B][/I][/U]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 [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Splitting Fields: Anderson and Feil, Theorem 45.4 ....
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