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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 an aspect of the proof of Theorem 45.6 ...
Theorem 45.6 and its proof read as follows:
https://www.physicsforums.com/attachments/6701At the start of the proof of Theorem 45.6 we read the following:
"Suppose that $$K$$ is a normal extension of $$F$$, a field with characteristic zero. Then by Theorem 45.5, $$K = F( \alpha )$$, where $$\alpha$$ is algebraic over $$F$$. ... .. "
Can someone please explain exactly how $$K = F( \alpha )$$ follows in the above statement ... ?The quote mentions Anderson and Feil's Theorem 45.5 and also mentions that $$K$$ is a normal extension so I am providing the statement of Theorem 45.5 and Anderson and Feil's definition of a normal extension as follows ... ...
View attachment 6702
https://www.physicsforums.com/attachments/6703
I am currently focused on Ch. 45: The Splitting Field ... ...
I need some help with an aspect of the proof of Theorem 45.6 ...
Theorem 45.6 and its proof read as follows:
https://www.physicsforums.com/attachments/6701At the start of the proof of Theorem 45.6 we read the following:
"Suppose that $$K$$ is a normal extension of $$F$$, a field with characteristic zero. Then by Theorem 45.5, $$K = F( \alpha )$$, where $$\alpha$$ is algebraic over $$F$$. ... .. "
Can someone please explain exactly how $$K = F( \alpha )$$ follows in the above statement ... ?The quote mentions Anderson and Feil's Theorem 45.5 and also mentions that $$K$$ is a normal extension so I am providing the statement of Theorem 45.5 and Anderson and Feil's definition of a normal extension as follows ... ...
View attachment 6702
https://www.physicsforums.com/attachments/6703