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I am reading Dummit and Foote, Chapter 13 - Field Theory.
I am currently studying Section 13.2 : Algebraic Extensions
I need some help with Exercise 13 of Section 13.2 ... ... indeed, I have not been able to make a meaningful start on the problem
Exercise 13 of Section 13.2 reads as follows:
View attachment 6611
Definitions that may be relevant to solving this exercise include the following:View attachment 6612A result which seems relevant is Lemma 16 plus the remarks that follow its proof ... Lemma 16 and the following remarks read as follows:
View attachment 6613
As indicated above I need help in order to make a meaningful or significant start on the solution to this exercise .. ...
One thought, though ... there must be some way to use $$\alpha_i^2 \in \mathbb{Q}$$ ... perhaps in establishing the dimension of $$F( \alpha_1 , \alpha_2, \ ... \ ... \ \ \alpha_{k + 1} )$$ over $$F( \alpha_1 , \alpha_2, \ ... \ ... \ \ \alpha_{k } )$$ ... and hence getting some knowledge of $$F( \alpha_1 , \alpha_2, \ ... \ ... \ \ \alpha_n )$$ over $$\mathbb{Q}$$ ... but even if we do gain such knowledge, how do we use it to show $$\sqrt [3]{2} \notin F$$ ... ...
... AND FURTHER ... anyway ... what is implied by $$\alpha_i^2 \in \mathbb{Q}$$ ... ... ?
Help will be much appreciated ...Peter
I am currently studying Section 13.2 : Algebraic Extensions
I need some help with Exercise 13 of Section 13.2 ... ... indeed, I have not been able to make a meaningful start on the problem
Exercise 13 of Section 13.2 reads as follows:
View attachment 6611
Definitions that may be relevant to solving this exercise include the following:View attachment 6612A result which seems relevant is Lemma 16 plus the remarks that follow its proof ... Lemma 16 and the following remarks read as follows:
View attachment 6613
As indicated above I need help in order to make a meaningful or significant start on the solution to this exercise .. ...
One thought, though ... there must be some way to use $$\alpha_i^2 \in \mathbb{Q}$$ ... perhaps in establishing the dimension of $$F( \alpha_1 , \alpha_2, \ ... \ ... \ \ \alpha_{k + 1} )$$ over $$F( \alpha_1 , \alpha_2, \ ... \ ... \ \ \alpha_{k } )$$ ... and hence getting some knowledge of $$F( \alpha_1 , \alpha_2, \ ... \ ... \ \ \alpha_n )$$ over $$\mathbb{Q}$$ ... but even if we do gain such knowledge, how do we use it to show $$\sqrt [3]{2} \notin F$$ ... ...
... AND FURTHER ... anyway ... what is implied by $$\alpha_i^2 \in \mathbb{Q}$$ ... ... ?
Help will be much appreciated ...Peter
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