Field Extensions - Lovett, Theorem 7.1.10 - Yet 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 yet another aspect of the proof of Theorem 7.1.10 ...Theorem 7.1.10, and its proof, reads as follows:
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I need help in fully understanding Lovett's argument that \(\displaystyle b(x)\) must be a constant (an argument I am having trouble fully understanding ... )

I have three questions ... ... (QUESTION 1) Lovett brings the minimal polynomial \(\displaystyle p(x)\) into the argument ... why is he doing this ... what is his objective in this matter ... ?Further ... ... Lovett writes ... :" ... ... Hence \(\displaystyle a( \alpha ) / b( \alpha )\) can be written as \(\displaystyle a_2( \alpha ) / b_2( \alpha ) \) where \(\displaystyle \text{ deg } b_2( x ) \lt \text{ deg } b( x)\). This contradicts the choice that \(\displaystyle b( x)\) has minimal degree. Consequently, \(\displaystyle r(x) = 0\) and hence \(\displaystyle b(x)\) divides \(\displaystyle a(x)\). ... .. "(QUESTION 2) I cannot see how in this argument Lovett concludes that \(\displaystyle b(x)\) divides \(\displaystyle a(x)\) ... ... ? Can someone please help ...?

Then Lovett writes:" ... ... Then the expression \(\displaystyle \gamma = a( \alpha ) / b( \alpha )\) can only be such that \(\displaystyle b(x)\) has minimal degree among such rational expressions if \(\displaystyle b(x)\) is a constant. ... ... "(QUESTION 3) I do not follow this argument that \(\displaystyle b(x)\) must be a constant ... can someone please help ... ?Peter

 

[jvicens]

, thank you for bringing up these questions and seeking help in understanding the proof of Theorem 7.1.10. Field extensions can be a challenging topic, so it's great that you are actively seeking to fully understand the material.

To answer your first question, Lovett brings in the minimal polynomial p(x) because it is the key to understanding the structure of the field extension. Remember that the minimal polynomial is the monic polynomial of smallest degree that has the root \alpha. In this case, we are considering the field extension \mathbb{Q}(\alpha) where \alpha is a root of p(x). Lovett's objective in bringing in p(x) is to show that the expression a(\alpha)/b(\alpha) can only be such that b(x) has minimal degree among such rational expressions if b(x) is a constant. This leads us to your third question.

In the proof, Lovett shows that if b(x) is not a constant, then there must exist another polynomial b_2(x) of smaller degree that can also be written as a(\alpha)/b(\alpha). This contradicts the choice that b(x) has minimal degree, so we can conclude that b(x) must be a constant. This constant can be factored out of the expression a(\alpha)/b(\alpha), leaving us with a polynomial expression that is equal to \gamma. Since \gamma is an arbitrary element of the field extension \mathbb{Q}(\alpha), we can conclude that b(x) divides a(x) for all elements in \mathbb{Q}(\alpha).

I hope this helps clarify the argument for you. If you have any further questions, please don't hesitate to ask. Good luck with your studies!