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I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 7: Field Extensions ... ...

I need help with another aspect of the proof of Theorem 7.1.10 ...Theorem 7.1.10, and its proof, reads as follows:

In the proof of the above Theorem, towards the end of the proof, Lovett concerns himself with proving that ##F( \alpha ) \subset F[ \alpha ]## ... ...

To do this he points out that every element in ##F( \alpha )## can be written as a rational expression of ##\alpha##, namely ...

##\gamma = \frac{ a( \alpha )}{ b( \alpha )}##

where ##a( \alpha ), b( \alpha ) \in F[x]## and ##b( \alpha ) \neq 0## ...

Lovett then says ...

" ... ... Suppose also that ##a( \alpha )## and ##b( \alpha )## are chosen such that ##b( \alpha )## has minimal degree and ##\gamma = \frac{ a( \alpha )}{ b( \alpha )}##. ... ... "What does Lovett mean by choosing ##a( \alpha )## and ##b( \alpha )## such that ##b( \alpha )## has minimal degree ... ... ?

It cannot mean choosing special elements for ##b( \alpha )## ... as then ##\gamma## would not be a representative element of ##F( \alpha )## ...

Can someone please clarify this issue ...

Peter

I am currently focused on Chapter 7: Field Extensions ... ...

I need help with another aspect of the proof of Theorem 7.1.10 ...Theorem 7.1.10, and its proof, reads as follows:

In the proof of the above Theorem, towards the end of the proof, Lovett concerns himself with proving that ##F( \alpha ) \subset F[ \alpha ]## ... ...

To do this he points out that every element in ##F( \alpha )## can be written as a rational expression of ##\alpha##, namely ...

##\gamma = \frac{ a( \alpha )}{ b( \alpha )}##

where ##a( \alpha ), b( \alpha ) \in F[x]## and ##b( \alpha ) \neq 0## ...

Lovett then says ...

" ... ... Suppose also that ##a( \alpha )## and ##b( \alpha )## are chosen such that ##b( \alpha )## has minimal degree and ##\gamma = \frac{ a( \alpha )}{ b( \alpha )}##. ... ... "What does Lovett mean by choosing ##a( \alpha )## and ##b( \alpha )## such that ##b( \alpha )## has minimal degree ... ... ?

It cannot mean choosing special elements for ##b( \alpha )## ... as then ##\gamma## would not be a representative element of ##F( \alpha )## ...

Can someone please clarify this issue ...

Peter

__Does it just mean that when we have ... for example ...##\gamma = \frac{ a( \alpha )}{ b( \alpha )} = \frac{ x^3 - 3x }{ x^4 + 7x }##we just (in this case, for example) we just 'cancel' the ##x## ... and similarly for other examples ...__**EDIT**