Field Extensions - Lovett, Theorem 7.1.10 - Another question

[Math Amateur]

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:https://www.physicsforums.com/attachments/6585
https://www.physicsforums.com/attachments/6586
In the proof of the above Theorem, towards the end of the proof, Lovett concerns himself with proving that \(\displaystyle F( \alpha ) \subset F[ \alpha ]\) ... ... To do this he points out that every element in \(\displaystyle F( \alpha )\) can be written as a rational expression of \(\displaystyle \alpha\), namely ... \(\displaystyle \gamma = \frac{ a( \alpha )}{ b( \alpha )}\) where \(\displaystyle a( \alpha ), b( \alpha ) \in F[x]\) and \(\displaystyle b( \alpha ) \neq 0\) ...Lovett then says ...

"suppose also that \(\displaystyle a( \alpha )\) and \(\displaystyle b( \alpha )\) are chosen such that \(\displaystyle b( \alpha )\) has minimal degree and \(\displaystyle \gamma = \frac{ a( \alpha )}{ b( \alpha )}\). ... ... "What does Lovett mean by choosing \(\displaystyle a( \alpha )\) and \(\displaystyle b( \alpha )\) such that \(\displaystyle b( \alpha )\) has minimal degree ... ... ?It cannot mean choosing special elements \(\displaystyle b( \alpha )\) ... as then \(\displaystyle \gamma\) would not be a representative element of \(\displaystyle F( \alpha )\) ...
Can someone please clarify this issue ...

Peter

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

 

[jvicens]

?Dear Peter,

Thank you for bringing up this question. In this context, choosing a( \alpha ) and b( \alpha ) such that b( \alpha ) has minimal degree means that we are choosing a representative element \gamma = \frac{ a( \alpha )}{ b( \alpha )} that is in its simplest form. This is similar to simplifying fractions in arithmetic, where we choose the smallest possible numbers to represent the fraction.

In the example you provided, \gamma = \frac{ x^3 - 3x }{ x^4 + 7x } can be simplified by factoring out an x from the numerator and denominator, resulting in \gamma = \frac{ x(x^2 - 3) }{ x(x^3 + 7) }. This is a simpler form of the fraction, as the degree of the denominator has been reduced.

So in general, when proving that F( \alpha ) \subset F[ \alpha ], we want to choose representative elements that are in their simplest form, with minimal degree for the denominator. This ensures that any element in F( \alpha ) can be written as a rational expression of \alpha, and thus belongs in F[ \alpha ].

I hope this clarifies the issue for you. Please let me know if you have any further questions or need additional clarification.