Splitting Fields and Separable Polynomials ....

[Math Amateur]

I am reading both David S. Dummit and Richard M. Foote : Abstract Algebra and Paul E. Bland's book: The Basics of Abstract Algebra ... ...

I am trying to understand separable polynomials ... ... but D&F and Bland seem to define them slightly differently and interpret the application of the definitions differently in examples ... I need help to understand why these things appear different and what the significance and implications of the differences are ...D&F define separable polynomial ... and give an example as follows:View attachment 6630
View attachment 6631
Bland defines separable polynomials as follows ... and also gives an example ...
View attachment 6632
https://www.physicsforums.com/attachments/6633
My questions are as follows:Question 1 Now ... for Bland, to qualify to be a separable polynomial, a polynomial must be irreducible ... and then it must have no non-distinct roots ...

For D&F any polynomial that has no multiple roots is separable ...Is this difference in definitions significant?

Which is the more usual definition?
Question 2In D&F in Example 1 we are given a polynomial \(\displaystyle f(x) = x^2 - 2\) as an example of a separable polynomial ...

... and ... D&F also as us to consider \(\displaystyle (x^2 - 2)^n\) for \(\displaystyle n \ge 2\) as inseparable as it has repeated or multiple roots \(\displaystyle \pm \sqrt{2}\) ...

... a particular case would be \(\displaystyle (x^2 - 2)^2\) and a similar analysis would mean \(\displaystyle (x^2 + 2)^2\) would also be inseparable ...BUT ...

Bland analyses the polynomial \(\displaystyle f(x) = (x^2 + 2)^2 ( x^2 - 3)\) and comes to the conclusion that f is separable ... when I think that D&Fs analysis would have found the polynomial to be inseparable ...

Can someone explain and reconcile the differences in D&F and Bland's approaches and solutions ... ...

Question 3In D&F Example 1 we read ...

" ... ... The polynomial \(\displaystyle x^2 - 2\) is separable over \(\displaystyle \mathbb{Q}\) ... ... "I am curious and somewhat puzzled and perplexed about how the term "over" applies to a separable polynomial ... both D&F and Bland define separability in terms of distinct or non-multiple roots ... they do not really define separability OVER something ...

Can someone explain how "over" comes into the definition and how a polynomial can be separable over one field but not separable over another ... ?

Hope that someone can help ...

Peter

 

[jvicens]

Dear Peter,

Thank you for reaching out for help with understanding separable polynomials. it is always important to carefully consider definitions and interpretations in order to fully understand a concept. I will address each of your questions in turn.

Question 1:
The difference in definitions between D&F and Bland is significant in that they are considering separability in different contexts. D&F is defining separability for a polynomial over any field, while Bland is considering separability specifically for polynomials over an algebraically closed field. In general, the more usual definition is the one given by D&F, as it is more general and can be applied to a wider range of fields.

Question 2:
The difference in analysis between D&F and Bland for the polynomial (x^2 + 2)^2 (x^2 - 3) is due to the fact that Bland is considering polynomials over an algebraically closed field, while D&F is considering polynomials over any field. In this case, (x^2 + 2)^2 has no multiple roots over an algebraically closed field, so Bland considers it to be separable. However, over any other field, (x^2 + 2)^2 has repeated roots and would be considered inseparable by D&F's definition.

Question 3:
The term "over" in this context refers to the field in which the polynomial is being considered. D&F is stating that x^2 - 2 is separable over the field of rational numbers, denoted by \mathbb{Q}. This means that when considering x^2 - 2 as a polynomial over \mathbb{Q}, it has distinct roots. However, if we were to consider x^2 - 2 as a polynomial over a different field, it may not have distinct roots and would therefore not be separable over that field. This concept is important in understanding the applications and implications of separable polynomials in different fields.

I hope this helps to clarify the differences in definitions and interpretations between D&F and Bland. It is important to carefully consider the context in which a concept is being applied in order to fully understand it. If you have any further questions, please do not hesitate to reach out.