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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 an aspect of Propositions 11 and 12 ... ...
Propositions 11 and 12 read as follows:
https://www.physicsforums.com/attachments/6606https://www.physicsforums.com/attachments/6607
Now Proposition 11 states that the degree of $$F( \alpha )$$ over $$F$$ is equal to the degree of the minimum polynomial ... ... that is
$$[ F( \alpha ) \ : \ F ] = \text{ deg } m_\alpha (x) = \text{ deg } \alpha
$$
... ... BUT ... ...... ... Proposition 12 states that ... "if $$\alpha$$ is an element of an extension of degree $$n$$ over $$F$$, then $$\alpha$$ satisfies a polynomial of degree at most $$n$$ over $$F$$ ... ... "Doesn't Proposition 11 guarantee that the polynomial (the minimum polynomial) is actually of degree equal to $$n$$?Can someone please explain in simple terms how these statements are consistent?Help will be appreciated ...
Peter
I am currently studying Section 13.2 : Algebraic Extensions
I need some help with an aspect of Propositions 11 and 12 ... ...
Propositions 11 and 12 read as follows:
https://www.physicsforums.com/attachments/6606https://www.physicsforums.com/attachments/6607
Now Proposition 11 states that the degree of $$F( \alpha )$$ over $$F$$ is equal to the degree of the minimum polynomial ... ... that is
$$[ F( \alpha ) \ : \ F ] = \text{ deg } m_\alpha (x) = \text{ deg } \alpha
$$
... ... BUT ... ...... ... Proposition 12 states that ... "if $$\alpha$$ is an element of an extension of degree $$n$$ over $$F$$, then $$\alpha$$ satisfies a polynomial of degree at most $$n$$ over $$F$$ ... ... "Doesn't Proposition 11 guarantee that the polynomial (the minimum polynomial) is actually of degree equal to $$n$$?Can someone please explain in simple terms how these statements are consistent?Help will be appreciated ...
Peter