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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:
Bland defines separable polynomials as follows ... and also gives an example ...
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 nondistinct 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 2
In D&F in Example 1 we are given a polynomial ##f(x) = x^2  2## as an example of a separable polynomial ...
... and ... D&F also as us to consider ##(x^2  2)^n## for ##n \ge 2## as inseparable as it has repeated or multiple roots ##\pm \sqrt{2}## ...
... a particular case would be ##(x^2  2)^2## and a similar analysis would mean ##(x^2 + 2)^2## would also be inseparable ...
BUT ...
Bland analyses the polynomial ##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 3
In D&F Example 1 we read ...
" ... ... The polynomial ##x^2  2## is separable over ##\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 nonmultiple 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
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:
Bland defines separable polynomials as follows ... and also gives an example ...
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 nondistinct 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 2
In D&F in Example 1 we are given a polynomial ##f(x) = x^2  2## as an example of a separable polynomial ...
... and ... D&F also as us to consider ##(x^2  2)^n## for ##n \ge 2## as inseparable as it has repeated or multiple roots ##\pm \sqrt{2}## ...
... a particular case would be ##(x^2  2)^2## and a similar analysis would mean ##(x^2 + 2)^2## would also be inseparable ...
BUT ...
Bland analyses the polynomial ##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 3
In D&F Example 1 we read ...
" ... ... The polynomial ##x^2  2## is separable over ##\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 nonmultiple 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
Attachments

D&F  PART 1  Definition of Separable Polynomial and Example  PART 1 ... ....png37.5 KB · Views: 414

D&F  Definition of Separable Polynomial and Example ... ....png63.5 KB · Views: 552

Bland  Definition of a Separable Polynomial ....png26.5 KB · Views: 586

Bland  Example of a Separable Polynomial ....png55.2 KB · Views: 464